ID 5138

*Computer Update Problem *

I spot a computer at work which is doing updates.

When I arrive it says '93% complete'.

I immediately start timing the activity and it takes 8 minutes to change to '95% complete'.

Assuming the 'percentage complete' relates directly to time, how much longer will it take to reach '100% complete'?

ID 5139

Eugenia wants to make a simple bridge for her dog. Currently he has to run through a tiny stream in the back garden and then walks mud into the house. Since Eugenia’s dad owns and runs a machine shop, she can easily get a single sheet of steel, aluminum or wood to bridge the stream.

The length suits the size of the stream, the width suits the size of the dog, and the weight will be as much as she can carry.

The strength of a plain sheet is proportional to the relative strength of the material, its width and the cube of its thickness.

Which of the available materials makes the strongest bridge?

ID 5143

Now that I have made a vast fortune from my patented premium dog biscuits, I can afford to build the luxury mansion of my dreams.

I thought my design requirement was very clear: The water for the walk-through shower can be turned ON and OFF from both ends of the room.

The plumber doesn't understand so I have drawn him a plan. I had a few attempts before I got it right!

Which is the correct drawing?

ID 5146

A plastic ruler is much easier to bend in one direction than another according to the proportionality shown, where **b** is the breadth and **d** is the depth of the rectangular cross-section.

The ruler can be approximated as being 1mm thick and 20mm wide.

How much stiffer is it in the hard-to-bend direction?

WARNING: Unless the ruler has some sort of anti-shatter statement written on it, do not try bending it like this without using eye protection. Older designs of rulers are known to shatter and eject bits of plastic into nearby eyes.

ID 5147

I roll two dice, one with the left hand and one with the right.

If the left hand die gives an odd number, the overall score is zero.

If the right hand die gives an even number, I roll it again and again until it is odd.

The score is the sum of the two numbers, except for the previously mentioned case.

There are exactly 6 possible scores: 0, 3, 5, 7, 9, and 11.

What is the probability of a score of 3?

ID 5148

I roll two dice, one with the left hand and one with the right.

If the left hand die gives an odd number, the overall score is zero.

If the right hand die gives an even number, I roll it again and again until it is odd.

The score is the sum of the two numbers, except for the previously mentioned case.

There are exactly 6 possible scores: 0, 3, 5, 7, 9, and 11.

What is the average score?

ID 5152

There are two gods named Orbis and Fidelis, one on your left, the other on your right, but you do not know which is which. Fidelis always answers correctly. Orbis only answers alternate questions correctly; you do not know if his last answer was correct.

You must determine which god is which using the minimum number of YES/NO questions. How many questions do you need to be certain?

To clarify the matter, if Orbis is answering incorrectly your entire question is evaluated correctly and then the answer is reversed. A single question to both gods counts as two questions.

Inspired by G. Boolos. 'The hardest logic puzzle ever', The Harvard Review of Philosophy (6), 1996.

ID 5164

I am allergic to washing powder, but I don’t want my shirt to smell. The dermatologist has told me to reduce the total amount of washing powder residue on my shirt to less than 1 pico-gram. (I think he just made that number up on the spot!)

My dry shirt measures 280g, but after washing and spin-drying it weighs 350g. The washing and rinsing uses 21 L of fresh water for each operation. I wash my shirt with one 30g tablet of washing powder.

How many times do I have to rinse my shirt to reach the required non-allergenic state?

(remember that the density of water is 1g / mL, 1000 mL = 1 L, 1 pico-gram = 10^{-12}g).

ID 5166

Sammy the sea lion has spotted a gorgeous Lady sea lion before anyone else. She is in the sea and he is on the beach. The picture shows the distances in hundreds meters. He needs to get to her as soon as possible to avoid any potential rivals getting to her first.

He can travel three times faster in the sea compared to on the sandy beach over the distances involved.

Which route should he use?

ID 5167

The stand-in mathematics teacher is forced to choose two students to go on a field trip and he can only choose between the two best girls and the two best boys in the class. He hates the idea of a girl being paired up with a boy, but knows that if he first picks a boy it is much more likely that the next pick will be a girl.

He devises this scheme: He labels 4 otherwise identical tokens with the names of the four students. He puts the tokens into a bag and then reaches in and takes two tokens at exactly the same time, one in each hand.

What is the chance of a boy being paired with a girl with this cunning plan?

ID 5168

Given Cartesian axes of infinite extent, find the ratio of the areas below to above the semi-infinite 45° inclined blue lines shown.

ID 5169

Find the ratio A/B where A is the number of elements in the set of all positive integers

and B is the number of elements in the set of all even positive integers.

ID 5170

Evguenia's birthday parties are always the same, and always boring. She gets seated at the table first. There are three other places set with named seating positions. Granddad always arrives next, and since he refuses to wear his glasses he always sits in a random position. Then her Aunt arrives, and if her place is free she takes it. Otherwise she sits in a random position. Then her mother arrives, and if her seat is taken she is grumpy all evening.

What is the probability that her mother gets to sit in her named seat?

ID 5171

A road traffic junction has no provision for pedestrians. The traffic lights are just changing to stop the RED and ORANGE cars. The GREEN and BLUE cars are next to go. Pedestrians A and B will only start to cross once all cars are stationary.

Compare the crossings for A and B.

(NOTE: Some jurisidictions allow drivers to pass red lights when turning, but that is not allowed at this crossing.)

ID 5173

Jane and Gerry work at a call center selling worthless rubbish to unsuspecting customers. They are in competition with each other to be the best seller today. In the morning they work from the *easy customer* list and in the afternoon they are forced to work from the *difficult customer* list.

In the morning Jane sells to 90 out of her 100 calls, whereas Gerry sells to 85 out of his 100 calls. 90% to 85% means Jane wins.

In the afternoon Jane sells to 30 out of her 100 calls, whereas Gerry, who is sad that Jane is winning, only makes 20 calls, and only makes 5 sales. With 30% to 25% sales figures, Jane again wins.

At the end of the day the boss totals the sales, totals the calls, and computes the aggregated percentage successful sales figure for each seller.

Who wins the competition?

ID 5175

The King of a far away land needs to adjust the temperature of the shower all by himself. The inconsiderate servant who normally performs this duty has broken his leg and is therefore unavailable. (The King has failed to realise that kicking the servant caused the servant to trip, breaking his leg because of the fall!)

The King requires a flow rate of 4 liters per minute of 40°C water to be derived from two calibrated hot and cold water taps. The cold water is at 10°C and the hot water is at 50°C. The mixing unit does not change the calibration of the taps or allow heat to be lost from the water.

What settings are required on the taps?

ID 5178

The King of a far off land has just had the ballroom lights in his palace rewired by probably the most illogical and incompetent electrician in the land. Instead of each of the 8 up/down switch positions turning on one of the banks of lights in the ballroom, each switch has to be in exactly the correct up/down state in order to make all the lights come on at once. In every other switch position all the lights are turned off.

Needless to say the incompetence of the electrician enraged the King so much that the electrician was executed in a gruesome way the same day.

Unfortunately, for some inexplicable reason, electricians seem reluctant to come and fix the problem. The Chief Steward needs to devise a plan to turn the lights on and off every evening so the King isn’t made to look foolish in front of his important guests over the coming weeks.

How many switch positions need to be changed every night to meet the requirements?

Changing any switch lever from down to up, for example, counts as one position change.

ID 5179

Modern petrol-electric hybrid cars typically have so many sensors that you can do interesting things like measure, log, and plot the power going to the wheels. This is such a plot, measured on level ground during the course of one day with relatively constant temperatures, no rain, and not especially windy.

Below 30 mph the car was running on electric power only. Above that it was doing its own hybrid thing with the engine running.

What can you conclude from the plot alone?

(NOTE: The power should at least double every time you double the speed since at twice the speed that part of the journey is half as long.)

ID 5180

Criminal gangs have been known to pay vagrants to search through people’s refuse to find useful information like bank statements, credit card statements, receipts and so forth. With such personal information the criminals can then pretend to be the householder and take out loans, buy things or do other bad things having stolen somebody’s identity.

Steve is fairly careful about shredding such documents, but about 5% of the time an important document slips into the refuse unshredded. Fortunately, unless he is being specifically targeted, it is pretty unlikely that somebody will be going through his refuse every week. Let’s put the odds at 1 in 1000 for each weekly collection.

Assuming that if Steve fails to shred a document, and if the criminals are searching his bins at that time, his identity will be stolen, what is the chance of that happening in a 10 year period?

ID 5209

It has been several years since the Apocalypse, but the Zombies still seem to be everywhere. I have been caught out in the open on my own and am now surrounded by 3 hungry Zombies, intent on eating my brains. Fortunately I have 6 rounds (“bullets”) in my gun. Unfortunately the ammo is old and degraded so it only works 80% of time. Also, although my aim is excellent on a shooting range, my shots are inaccurate when I am nervous, for example when I am surrounded by Zombies! It turns out that the closer they get, the more nervous I get, so the chance of my getting a shot to their head is only 63%. (Everybody knows that only a shot to the head will kill a Zombie). I worked it out, there is roughly 50% chance that any particular round will end up killing a Zombie.

There is just enough time to fire off all 6 rounds.

What is my chance of living to fight another day?

ID 5210

The more debt you get into, the more credit card companies profit. If you buy $100 of goods at a shop you owe the credit card company $100, but typically the shop only gets $98. The shop has to inflate its prices to pay the credit card company. Typically the shop is contractually not allowed to give a discount for cash.

The CostCrashers supermarket chain deals only in cash. The CardPayers supermarkets deals both with both cash and credit card transactions, although 75% of people pay by credit card.

All else being equal, how much cheaper could the CostCrashers prices be compared to the CardPayers supermarkets, given the figures stated earlier?

ID 5212

It is difficult to see a 2% price difference between food items in two different food stores, but the difference becomes noticeable over time.

If the weekly family shopping bill is $200, how much money does a 2% reduction make over the course of a year.

ID 5213

Shops have sales all the time to attract your business. Let the buyer beware! Not all sales are as good as others.

Given the same branded goods being sold, and the same quality of after-sales service, which shop offers the best value, given that two weeks ago all had the same price.

ID 5214

Income tax is a strange concept when inflation is taken into account.

At the end of one year with a 1% interest rate and a 2% rate of inflation, what is the tax owed on a $10,000 saved amount, given a 20% income tax rate?

ID 5215

Some forms of loans are more *iniquitous* than others. Find the loan method with the most extortionate rate.

*(Definition: Iniquitous – grossly unfair and morally wrong) *

NOTE: in some regions, Lenders are required to state the Annual Percentage Rate (APR) in order to make comparisons easier.

ID 5216

Jane was a naughty little girl. When she used to play 2-dice games with her late grandfather she always used to cheat. Her grandfather would pretend not to notice that the dice had landed and that she quickly changed one of the dice to her advantage. Specifically, if it was her throw she changed the lowest die to a 6. If it was his throw she changed the highest die to a 1.

Over a long run of throws, how much bigger was Jane’s average score than her grandfather’s?

ID 5217

My Interocitor is broken. Fortunately I have replacements for every single part and each part takes the same time to fit as any other. I also know that the designers were so unbelievably clever that it is inconceivable that one faulty part would cause other parts to fail.

The Interocitor has 60 parts, all of which are different, and I can change from one part to another in 10 seconds without even turning the power off (*hot-swappable parts*). I will know that it is working immediately.

What is the minimum time in which I can guarantee to have fixed it?

An interocitor is a fictional multi-functional device that first appeared in the 1949 story "The Alien Machine".

ID 5218

A certain Professor of Statistics is trying to explain to his grandson that once an event has happened its probability of happening is 1 because it is certain that it happened. The grandson disagrees.

The grandson throws a pair of dice and as they are about to settle puts a bowl over them, preventing the Professor from seeing the result. The grandson then peeks at the dice, knowing the value. He then explains that he could now show these dice to an infinite number of people, other than the professor, and yet the professor still could not declare with certainty what the result was, despite the fact that on average, everybody knew!

What can we say with certainty about this situation?

ID 5221

My neighbor John has invented a perpetual motion machine. It pumps water with no apparent power input and can even pump water up over a 2m fence.

How would you categorise this invention?

ID 5222

What is the next number in the sequence?

**0, 3, 1, 4, 2 **

(NOTE: It is a single sequence and not two sequences interleaved.)

ID 5223

Three observers, not more than 1km apart on flat windless terrain, report the same event quite differently.

They meet up later. Amy says the low frequency sound occurred first, followed by the high frequency sound about 1 second later. John says she is an idiot and claims the high frequency sound happened first, followed by the low frequency sound. Their teacher recorded the event and can prove that both sounds occurred within 0.01 seconds of each other, but being a kind teacher doesn’t actually tell them they are both stupid.

Which possible true statement demonstrates the least incompetence in the observers. . .

(HINT: the speed of sound does not change significantly for the frequencies heard, and there are no temperature gradients or fog to consider.)

ID 5225

"My neighbor a famous Professor of Physics has an excessively old and doddery gardener who knows absolutely nothing about basic science. Nevertheless this disrespectful old gardener insists that the Professor does not need to install an electric power cable all the way down to the ponds to power a fountain pump. The decrepit gardener insists that he has seen such a setup when he was a child and that it required no external power, had no electric motor, and could spray water well above the height of the source pond - but has no idea how. The Professor is still listening because the power cable will cost thousands of dollars to install.

What should he do?"

ID 5227

"Consider which would hurt less if it was accidentally dropped on your foot, a 30kg bag of cacao beans or a 30kg bag of feathers."

ID 5228

"** Endocrinic Igorosis** is a horrible disease with a 100% mortality rate for the infected. Even after 5 years nobody knows how it spreads. The fatality statistics have been steady at 100 people per 100,000 of the population every year. A remarkable cure has been found which is 100% effective at curing the disease, but only if administered before any symptoms are visible. Sadly, if the cure is given to anybody who is not infected then 3% of them will die. The latest test is 100% effective at finding this awful disease, but has a false positive rate of 5%.

How do we save the greatest number of people?"

ID 5229

"You accidentally knock an almost full opened can of soft drink over onto a carpeted floor.

Roughly how long have you got to make the can upright before over half of the contents are spilled?"

ID 5230

"Henrietta is threading the last bead onto a necklace when Tabby the cat brushes past the bowls of undrilled beads, knocking them over.

500 rondelle beads, 400 square rondelle beads, and the teardrop beads all end up on the work bench. Being startled by the falling beads, she drops the last bead onto the bench. There are now 1000 beads all mixed up on the bench and Henrietta need to find the one drilled bead she just dropped amongst all the undrilled beads.

Henrietta can't see the drilled hole without using the magnifier - which is on another bench. The drilled bead is neither a rondelle nor a square rondelle.

What is the chance she will pick the drilled bead on the first attempt?"

ID 5231

In a particular city it is illegal for those aged over 18 to consume children's chocolate drinks. Jack is walking home and drinking a chocolate drink. Sadly he is now 3 months too old to do so. When he sees a cop he therefore runs away and dumps the drink over a fence. The cop arrests him and charges him with arson. **WHAT!** Jack was wearing a green coat and the arsonist was seen wearing a green coat. At the police station the cops claim that only 2,000 people in this city of 100,000 people wear green coats. The odds of randomly finding him were therefore 2,000/100,000 making it 98% certain that he is the guilty party.

Using the evidence presented, what is the correct conclusion?

ID 5234

There are two gods named Mendax and Fidelis, one on your left, the other on your right, but you do not know which is which. Fidelis always answers correctly. Mendax only answers one question in 7 correctly in a repeating cycle. You do not know which part of the cycle Mendax is currently on.

You must determine which god is which using the minimum number of YES/NO questions. How many questions do you need to be certain?

To clarify the matter, if Mendax is answering incorrectly, your entire question is evaluated correctly and then the answer is reversed. A single question to both gods counts as two questions.

Inspired by G. Boolos. 'The hardest logic puzzle ever', The Harvard Review of Philosophy (6), 1996.

ID 5236

There are three goddesses named Mendax, Fidelis and Furtibus; one on your left, one on your right, and one in front. You do not know which is which. Fidelis always answers correctly. Mendax only answers falsely. Furtibus always answers in a way intended to best hide its identity.

You must determine which goddess is which using the minimum number of YES/NO questions. Each question is heard and answered by all three goddesses. That counts as one question.

How many questions do you need to ask in order to be certain?

[You should ideally think up actual questions before answering!]

Inspired by G. Boolos. 'The hardest logic puzzle ever', The Harvard Review of Philosophy (6), 1996.

ID 5238

"Suppose a particular nuclear waste material has a half-life of 100 years.

What could you do to reduce the radioactivity of the material itself to less than 7% of its current value?

(The half-life of a radioactive material is the time it takes, on average, for half of it to change into something else by spontaneous radioactive decay.)"

ID 5245

Dieter's new design of frequency-doubling power converter has an efficiency of 40%, a good figure for this particular type of device.

Kerstin's design is half the price so it is worthy of consideration, despite the fact that for the same power input, Kerstin's design produces 25% less output power than Dieter's.

What is the efficiency of Kerstin's design?

[ NOTE: Efficiency = 100% x (output power) / (input power) ]

ID 5252

* A phrase you will hear on the news or from people speaking is "the vast majority of". As a silly example you might hear something like "The vast majority of people with big noses also have big ears." What is the mathematical definition of the phrase "the vast majority of"?*

* There is a fault with the cruise control on Hank's car such that the speed continuously and linearly increases with time. When he starts off the speed is set to exactly 60 mph. He is driving on a long straight route with the radio on at full blast and he is not paying any attention to his speed. After 3 hours he notices that his speed has now reached 80 mph. How far did he travel in the first 3 hours? *

* There is a fault with the cruise control on Hank's car such that the speed continuously and linearly increases with time. When he starts off the speed is set to exactly 60 mph. He is driving on a long straight route with the radio on at full blast and he is not paying any attention to his speed. After 3 hours he notices that his speed has now reached 80 mph. For how many miles did he drive above the state speed limit of 70 mph? *

*Cletus is absolutely, definitely, the worst student the driving school has ever seen. When asked to drive at a steady speed he constantly accelerates too hard, overshoots the target speed, and then brakes too heavily. He has what the instructors call a "heavy foot". By some miracle he manages to always hit the same top speed of 20 mph before braking in a continuing non-repeating pattern like the one shown. What is his average speed? *

*In order to test the Archimedes Principle, Sasha put a Lead weight inside a football and makes it air-tight with glue. The football now weighs 500g. Sasha fills an outdoor water butt to the brim with water, but can't do the experiment that day because he has forgotten his gloves and the air temperature is 20°F. After a few days the temperature has not risen, but he finds his gloves and does the experiment. How much water is displaced by the football? If you measure the temperature in Celsius, there is a formula: The temperature T in degrees Celsius (°C) is equal to the temperature T in degrees Fahrenheit (°F) minus 32, times 5/9. *

*Sasha is learning about Archimedes at school. His show-off sister is 3 years older than him and sets him a problem: A water butt is filled to the brim with water and then tilted to a 30° angle relative to the horizontal. A football has a lead weight sealed inside, but not fixed in position. The combined weight of the ball and weight is 330g. The ball floats to a depth of 1.2 inches. Knowing the rough size of a football, estimate how much water spills out of the butt when the ball is lowered into the water gently. *

*The unit price of a thingamajig is $3.50. If I buy 10, the unit price drops to $2.00. At what quantity does it become more costly to buy them singly than to just buy 10? (Thingamajig is something whose name you have forgotten or do not know.) *

*The chocolate biscuit factory you are now in charge of has a problem. There are three 8 hour shifts and 11 shift supervisors who each have their own treasured setup of the machines to give an optimum biscuit pass rate. Every time the controls are adjusted the pass rate drops for several hours until the process settles down again. Each supervisor adjusts the controls to their “optimum” settings when their shift starts! Having analysed what they are doing, you have summarized the settings into 11 controls with two positions each. You need to devise a series of experiments to establish the optimum settings in a convincing manner to improve the productivity of your plant. What is the minimum number of experiments necessary to find the optimum settings for each control? *

*When it is 11am in Geneva (Switzerland) it is 5am in New York (USA). The flying time from Geneva to New York is 9 hours. If the plane to New York takes off from Geneva at 9:35am, what time is it in New York when the plane lands? (There are no stops or disruptions, just a normal flight.) *

*The train driver knows there is a Granny on board so he wants to give her a comfortable journey and to make sure she doesn’t spill her tea. When he is braking, which curve should he follow? (HINT: remember Newton’s Second Law, often expressed as Force = Mass x Acceleration) *

*You may have heard the expression "The shortest distance between two points is a straight line." I have drawn a straight line on the map, directly along one of the grid lines of the map. Is this the shortest path for a ship to travel? (Neglecting winds, tides and so on.) (NOTE: this is for a ship, not a submarine!) *

*Bethany throws a hard ball at John when they are in the park and it hurts. When they are weightless in the space station she somehow manages to throw the same ball at the same speed at John. What is the result?*

* Electrical energy can be measured in Joules, but for household use it is typically measured in kilowatt-hours (kWh). If a 100 Watt bulb is switched on for 8 hours every day for a year, roughly how much energy is consumed? J = 1000 x kW x S where J = Joules, kW = kilowatts, and S = seconds *

* Scientists and engineers need to use quantities which can vary over at least 30 orders of magnitude. Using zeros such as 0.000000000023 is impractical. Scientific notation uses a number between 1 and just less than 10, multiplied by a power of 10. What is the scientific form for 0.000123?*

* Scientists and engineers need to use quantities which can vary over at least 30 orders of magnitude. Using zeros such as 12300000000000 is impractical. Scientific notation uses a number between 1 and just less than 10, multiplied by a power of 10. What is the scientific form for 567001?*

* Engineers need to use quantities which can vary over at least 30 orders of magnitude. Using zeros such as 12300000000000 is impractical. Engineering notation uses a number between 1 and just less than 1000 followed by a standard multiplier for thousands, millions, and so forth. Express the frequency of 3 thousand million Hertz (Hz) in engineering notation.*

* Suppose the light bulb in your fridge uses 1 Watt when on and you pay 10 cents = $0.1 per kWh. Estimate the cost due to a faulty door switch, which keeps the light on over a 10 year period even with the fridge door closed. (1 kWh is 1000 Watts for 1 hour)*

* Peter changes his old fashioned single 100W incandescent light bulb to a super efficient 4 LED bulb system which gives the same light output, but only consumes 20W in total. Estimate the electrical energy saving per year if the bulbs are switched on for 8 hours every day of the year. (1 kWh is 1000 Watts for 1 hour)*

* Peter changes his old fashioned single 100W incandescent light bulb to a super efficient 4 LED bulb system which gives the same light output, but only consumes 20W in total. Estimate the saving per year if the bulbs are switched on for 8 hours every day of the year and you pay $0.1 per kWh. (1 kWh is 1000 Watts for 1 hour)*

*A professional design standard requires that free standing equipment must not tip over if subjected to a force equivalent to one fifth of its weight applied at the worst possible point. The latest design has failed the test. What can the design team do to fix the problem?*

*Answer the traditional English-language nursery rhyme in the form of a riddle: "As I was going to St Ives I met a man with seven wives Each wife carried seven sacks Each sack carried seven cats Cats, sacks, man, and wives How many were going to St Ives?" Leslie Green commented: "This is a variant of a traditional old rhyming puzzle, often interpreted as a trick question by assuming that the group was travelling *

*My neighbor Timmy has developed a complicated set of rules for choosing girlfriends. He likes: all girls with long hair; all girls who are not tall; tall girls who wear glasses; short-haired girls who don't wear glasses. Can you simplify the rules without changing the requirements?*

*In a large village community there are regularly women giving birth to triplets, but surprisingly never twins. The average number of babies per mother is 1.9 The local nurse visits the next recent mother on her list. Which is the most probable number of babies she will find?*

*Einstein, the hyper-intelligent house cat, is busy walking from one corner of a rectangular room to the diagonally opposite corner in an apparently random manner. His rather stupid human slaves can’t work out what he is doing. The light and dark carpet tiles on the floor look like part of a chess board, with exactly 10 tiles down the length and 6 tiles across the width. Einstein has decided to move from one corner of the room to the diagonally opposite corner, at each tile moving only down the length or across the width of the room. How many different routes are there from one corner to the diagonally opposite corner?
[HINT: You could try an easier problem first.] *

*Anton, the highest IQ house ant on the planet, is taking his regular nocturnal walk from one corner of the chess board (1,1) to the diagonally opposite corner (8,8), moving only right or up the board at each successive square. He remembers that on square (5,3) there is a 'black-hole', so any valid route must exclude this square. From how many different paths can he choose? [HINT: You could try an easier problem first.] *

*Anton, the highest IQ house ant on the planet, is taking his regular nocturnal walk from one corner of the chess board (1,1) to the diagonally opposite corner (8,8), moving only right or up the board at each successive square. He remembers that on square (5,3) there is a tasty sticky residue, so any valid route must involve this square. From how many different paths can he choose? [HINT: You could try an easier problem first.] *

*Martin, the mathematical mole, has dug an extensive network of underground tunnels which he has approximated, in his magnificent mole mind, as a 3D lattice of 30 x 30 x 30 intersections. The distance between intersections is approximately constant. He is currently at intersection (20, 20, 15) and wishes to get to his secret food cache at (25, 25, 20) by one of the many shortest routes available. There are stones blocking the intersections at (21, 22, 10), (22,19,18), (22, 22, 16), (24, 18, 18) and (26, 26, 18). From how many equally short paths can he choose? [HINT: You could try an easier problem first.] *

*Marek, the pan-dimensional super being, has arbitrarily defined his current location as (0,0,0,0,0,0) in 6D hyperspace. He wishes to reach location (3, 0, 2, 0, 4, 3) by one of the many shortest paths available. Despite his immense power, he can only move one hyperstep at a time, each hyperstep consisting of a unit change in exactly one of the coordinate values. Any hyperstep is of equal ‘length’. What is the smallest number of hypersteps required for him to reach his destination? *

*Marek, the pan-dimensional super being, has arbitrarily defined his current location as (0,0,0,0,0,0) in 6D hyperspace. He wishes to reach location (3, 0, 2, 0, 4, 3) by one of the many shortest paths available. Despite his immense power, he can only move one hyperstep at a time, each hyperstep consisting of a unit change in exactly one of the coordinate values. Any hyperstep is of equal ‘length’. From how many of the shortest paths can he choose? [HINT: You could try an easier problem first.] *

*A team of archaeologists is exploring an underground complex on a remote planet. On each level there is a regular grid of North-South corridors intersecting East-West corridors, with ladders at each junction going both up and down to the next levels. Effectively the complex appears to be a regular 3D lattice of tunnels. The previous team has marked the tunnels and made a list of problematic junctions that need to be avoided. The team is currently at junction (3, 2, 5) and needs to get to junction (12, 9, 8) by one of the many shortest available routes. Which of the listed problematic junctions might be in their way? *

*You are pitching your new idea to a panel of Venture Capitalists (VCs) to secure increased funding. Using your advanced mathematical skills, you have dumbed-down the probability of success to something even VCs can understand. You tell them that if they were to throw 10 normal dice and sum the dots on top, the probability of your success is the same as the sum being less than 50. One of the VCs seems very antagonistic, but you must still give the best possible answer, quickly – and using only mental arithmetic. His question is "Can you guarantee that the sum of dots would be less than 50?" *

*Shakuntala Devi was undoubtedly the most brilliant arithmetic mental calculator of all time. In 1977 she mentally calculated the 23 ^{rd} root of a 201 digit number in a mere 50 seconds. She toured the world showing how she could do calculations faster than they could be entered into and solved by the computers of the day. The problem for you is much simpler: Evaluate (without using a calculator) the 6^{th} root of the 25 digit number consisting of 1 followed by all zeros. *

*Shakuntala Devi was undoubtedly the most brilliant arithmetic mental calculator of all time. In 1977 she mentally calculated the 23 ^{rd} root of a 201 digit number in a mere 50 seconds. She toured the world showing how she could do calculations faster than they could be entered into and solved by the computers of the day. The problem for you is much simpler: Evaluate (without using a calculator) the 20^{th} root of the 11 digit number consisting of 1 followed by all zeros. *

*Symmetry is a very big subject, involving much more than geometry alone. Spotting patterns, and breaks in patterns, is a valuable skill. Without worrying about what the function is, or what the programming language is, can you spot the error in this code simply by spotting a break in the symmetry? The error is on line . . . *

* The sinusoidal waveform shown has a peak amplitude of 20 and a period of 2. What is a rough estimate for the mean value of the waveform over the interval shown (from t=0 to t=5)? (There is no need to use Calculus). HINT: Areas below the x-axis should be considered as negative when calculating the mean. *

*The boss of a 10 person company is always the last to arrive, and gets the worst of the 10 car parking places as a result. Being the boss, he decides to allocate parking places to each of his 9 employees (all of whom use a car to get to work), obviously keeping the best parking place for himself. The employees already hate the boss, who only got the job by marrying the owner. Further enraged by the new rule, they collectively decide to ignore it and just park randomly when they arrive. All places are good for the employees. What is the chance that the boss gets to park in the best parking place?*

*Jake, being bored on a rainy Sunday afternoon, throws a pair of dice 500 times and keeps a record of the results. What is the ratio of probabilities between throwing one five and all the rest twos, compared to throwing all threes. *

*An escaped criminal has stolen a spaceship, and has just instantaneously jumped 1 light year away. On each successive jump she will only be able to jump half the distance of the immediately preceding jump due to heat build-up. Jump engines always take 1 hour to recharge. My ship can only jump 1/2 light year, but it has a better cooling system so the jump distances drop-off more slowly. My maximum jump distances follow the sequence: 1/2 light year, 1/3 light year, 1/4 light year, and so on. I can find her with my subspace-tracker. If she is closer than my maximum jump distance I can get close enough to remotely disable her jump-drive and capture her. My jump engines are fully charged. Can I catch her? *

*Despite advice to the contrary from his friends and parents, Timmy has decided on a new strategy to select future girlfriends. He has two “must-see” programs on 5 days of each week. He requires that any future girlfriend must match-up with at least 90% of these programs. Given that there are 20 TV channels available in his area, what is the probability of a match? *

*There are 4 playing cards face up in a line on a table. Each of the cards has a different value. The cards need to be sorted so that the smallest value is on the left. There is only one action you can perform, namely interchanging the position of two cards (swapping them). You cannot move a card to an empty space. What is the minimal number of swaps achievable on the worst possible arrangement of cards? *

*The playing cards shown need to be sorted into increasing order with the lowest card on the left. Each move consists of picking up a card and inserting it anywhere in the line, including at the beginning or at the end. The cards slide sideways to allow a card to fit in between if necessary. Which is the fewest number of moves possible? *

*Christmas is getting near, and a four person company wants to run a Secret Santa scheme. The idea is that all four names are put into a hat and drawn at random. You buy a present for the person whose name you pick. Sadly, for the last three years in a row, at least one person has picked themselves, ruining the draw. Estimate the probability that the draw will fail this year because somebody picks themselves. *

*John refused to learn how to cook, despite the best efforts of his parents and teachers. Now he is at college, any meals he prepares can contain only some combination of boiled eggs, baked beans, and pizza. How many different meals can he prepare?*

*Jasmine has just been learning about the binary number system at school. On her way home she wondered how far she could count using just the four fingers on one hand, if a curled finger represented a binary 0 and an outstretched finger represented a binary 1. To be clear, she was thinking about counting up from zero in whole numbers. How far could she get? *

*The teacher walked into her classroom to find a scene of devastation. There was red paint on the walls, her lunch had been half eaten, and books were thrown around the room. There were only three children in the room: Alex, Betty, and Clive. All three said that Betty ate the lunch. Betty said Alex painted the wall. Clive said that Alex threw the books. Alex said that Clive painted the walls. The Headmaster was called in to resolve the crisis. On his way to the scene he found Wesley hiding in the corridor. Whilst Wesley would not directly implicate anyone, he did admit (under duress) that each of the three had done one of the crimes, and that every statement they made had been untrue. Given that Wesley is telling the truth, who threw the books? *

*A pirate captain takes his pirate crew into the treasure cave shown. The Captain marks the sturdy plank in units of the depth of a pirate, meaning pirates can stand on the marked positions, but not any closer. All of the pirates, including the Captain, are the same weight, and the bag of Gold is one quarter the weight of a pirate. The Captain, being bold and fearless, is going to walk on the unsecured plank over the edge of the steep cliff to position 8 and collect the Gold. He does not worry about falling to his death in the bottomless pit below. How many pirates are needed to stand on the plank to support the Captain? *

*Mark, who has been a digital design engineer for many years, is celebrating his 55 ^{th} birthday. Since the use of 55 candles for a birthday cake seems excessive, he arranges 7 candles in a line and lights the appropriate candles to represent his age in binary, a lit candle representing a “1” state for that bit. How many candles does he light? *

*Jenny is a computer scientist and is shy about her age. On her birthday she encodes her age in binary in a row of 8 candles. Her boyfriend, who is sitting on the opposite side of the cake, is trying to work out her age from the pattern of lit candles. Knowing that her boyfriend is fluent in binary, Jenny encodes the pattern correctly, but does not reveal if a lit candle represents a “1” or a “0” for that bit position. She also does not reveal if she has written the binary number either left to right increasing (standard notation) or the other way around. Given that Jenny is 27 years old, which age cannot reasonably be read by the boyfriend? *

*Jane would ordinarily like to drive at 60 mph on this particular stretch of road. However there is a large truck driving at 30 mph which is slowing her down. She knows that in 1 mile there will be a multi-lane road section (dual carriageway) where she can easily overtake, so she waits behind the truck. Gerry is impatient and overtakes both Jane and the truck to travel at 60 mph. Roughly how much time does Gerry save in this situation? *

*In the game of noughts and crosses (tic-tac-toe) the winner is the player who gets three of their symbols in a straight line, with each player placing their symbol alternately. In this game, *

*John accidentally drops his text book and it falls open at a random position somewhere near the middle of the book. He immediately counts the sum of the two visible page numbers. What is the probability that the sum of the *

**The History of Art in the Dark Ages** is an epically boring subject and too many students pass by simply answering the 100 multiple choice exam questions randomly. This year the marking scheme has been changed so that of the 4 possible answers, the correct answer scores one point, two wrong answers score 0, but the remaining stupid answer scores minus two points.

What is the expected score for a student who guesses randomly?

**The History of Art in the Dark Ages** is an epically boring subject and too many students pass by simply answering the 100 multiple choice exam questions randomly. This year the marking scheme has been changed so that of the 4 possible answers, the correct answer scores one point, two wrong answers score 0, but the remaining stupid answer scores minus two points.

What is the optimum strategy for a student who is good at Mathematics, but not Art History?

*My dog Charlie rushes into my office and bumps with his nose into a book on the floor at the position shown by the blue arrow. Describe what happens.*

*The contrapositive of a logical statement is formed by negating both the test and the result and then changing their order. For example: If [this is my house ] then [the door is black]. Becomes If [the door is not black] then [this is not my house]. Which is a correct contrapositive of If [this is a fish] then [it cannot live out of the water for very long]. *

*Johnny is a very poor communicator and a very fussy eater. If given a plate with any foodstuff that he doesn't like, he rejects the whole plate and sulks. He rejects 'spam, pizza & chips'. He accepts 'sausage, mash & spam'. He accepts 'pie, spam, & beans'. He accepts 'egg, chips, & spam'. He rejects 'spam, beans, & pizza'. Which food item is he actually rejecting? *

*It is demonstrable by direct computer calculation that any even number greater than 5 can be formed as the sum of two odd prime numbers. Find one of the two primes that sums to 36. *

*Analyze the following statement as if it were true: "HyperBrite cleans off 4x as much dirt as the nearest competitor." What can you say with *

*Since the year 2058, 11 year olds have been required to get high exam marks in one of four elective subjects in order to graduate; harder subjects can earn more points. The maximum possible scores are 100 for Set Theory, 200 for Vector Calculus, 300 for Orbital Mechanics, and 500 for Quantum Cryptography. Typically, students of Set Theory get 95 out of 100 questions correct in the exam, whereas the figures for the other subjects are 25 out of 50 for Vector Calculus, 11 out of 30 for Orbital Mechanics, and 21 out of 100 for Quantum Cryptography. Which subject gives a typical student the highest score? *

* Box A contains 3 blue balls and 1 red ball, all of the same size, weight, and texture. Box B contains 1 blue ball and 2 red balls, all indistinguishable from those in box A. I draw one ball from box A at random, examine it carefully, and put it into box B. I shake box B to mix up the balls, then draw a ball from that box at random. What is the probability that the ball is blue? *

*Two identical airplanes (aeroplanes) set out on a vital mission. The lead plane is carrying a secret message which needs to be delivered by hand. Each plane has a full fuel tank and a 1000 mile range. The planes can transfer fuel in mid-air; this process loses no fuel and happens almost instantly. How far can the lead plane get? (Note that a plane with no fuel can still land safely.) *

*Three identical airplanes (aeroplanes) set out on a vital mission. The lead plane is carrying a secret message which needs to be delivered by hand. Each plane has a full fuel tank and a 1200 mile range. The planes can transfer fuel in mid-air; this process loses no fuel and happens almost instantly. How far can the lead plane get? (Note that a plane with no fuel can still land safely.) *

*The image shows two parabolas, f(x)= x*

*What is the ratio of sides of a circumscribed regular hexagon to an inscribed regular hexagon sharing the same circle (as shown in the picture)? *

*The mathematics department at a school has challenged the pupils to a sort of ‘tug of war’ contest. The staff pull in the direction marked M. The boy students pull in the direction marked B, and the girls of course are G. The ropes are connected together by a fairly large strong equilateral triangular plate. The ropes are attached to the plate on smooth posts so the ropes are free to swing around the pivot. The relative strengths of the pulls and the position of the plate are shown at a particular moment in time. What happens next? *

*Johnny had a lesson at school which explained that ropes and cables are strong under tension but useless under compression. Brick and masonry, on the other hand, are strong under compression but weak under tension. Johnny has made the model suspension bridge (shown in the picture) using string and wooden blocks as a school project. Comment on the design. (Hint: consider the force on the top blocks) *

*When only one small uniform solid cylinder of ice floats (with its axis vertical) in a glass of cold fresh water then 8.3% of the length of the cylinder sticks out of the water, the rest being submerged. Estimate the specific gravity of ice. (Reminder: Specific gravity is the ratio of the density of something relative to the density of cold pure water.) HINT: Archimedes.
*

*You have measured the diagonal of your rectangular lawn using a laser rangefinder onto a conveniently positioned garden gnome. You have also measured the angle of the diagonal. What is the length of the lawn?
*

*Susan holds a pencil completely underwater at a 45° angle to the horizontal plane. Pencils have an average density less than that of water. What happens when she carefully releases the pencil?
*

*A drop of paint falls onto a horizontal flat sheet of clean glass. We suppose that at a particular instant the drop forms a perfect sphere in the air. The paint has spread out into a uniform circular disc (disk) of a diameter that is twice as large as the initial sphere diameter. What is the ratio of the disc thickness, *

*We are going to model the growth of a particular type of bacteria as follows: No growth below 3°C. Doubles in quantity every 20 minutes for temperatures between 10°C and 60°C. Dead at 80°C.
Compare the amount of dead bacteria in two similar portions of food, one of which is left to stand at room temperature for 2 hours, then quickly heated to 80°C, whilst the other portion is immediately heated to 80°C. The initial bacteria populations in the two portions are similar at the start of the day. *

*The teacher likes to reward students for being smart. This year the top three students each get to pick 10 cacao beans from a large bag. Each student is blindfolded and wears a glove to do the selection. Beans are removed one at a time, inspected, then replaced, with the bag contents being thoroughly mixed before the next pick. The bag contains hundreds of fresh beans, and an equal number of already dried beans. The student who gets the most dried beans wins a big prize. How does the best student optimise his chance of winning the prize? *

*This incident occurred in deep space. A space ship had been blasting its rockets at full power for several hours, such that the on-board accelerometers recorded an acceleration of 1g, the Earth normal gravitational acceleration of around 10m/s ^{2}. The Doppler Space Radar showed another space ship on a direct collision course so the Captain immediately cut off the engines. At this instant the other ship was 1000km away and a collision would happen in 1 hour if nothing was done. What can you say with certainty about the speed of the other space ship? *

*Terry the termite is taking a walk across the gap between two roses using a conveniently available cotton thread. Terry is quite good at mathematics, but not nearly as smart as his uncle Huygens. Uncle Huygens explained that the thread forms a shape known as a catenary, a curve which looks a bit like a parabola, but is more complicated than that. Terry is smart, but not that smart, so he approximates the curve as a 30° arc of a circle of radius 100mm. What is Terry’s estimate of the (arc) length of this cotton bridge?
*

*You are in a blue car on the side of the road. There is a lot of fast moving traffic going your way and you need to judge when to pull out into the traffic flow. If you want to wait for a really big gap in the traffic you could be stuck there for hours, so you need to pick a safe gap, but not an ideal huge gap. The traffic is going at an estimated speed of 56 mph (25 m/s). You are confident that you can get your car to accelerate from stationary to 25 m/s in 10 seconds (with constant acceleration). Measuring the gap between the rear of your car and the front of the car behind, what is the minimum gap that would not cause an accident (we assume that the other driver does not brake)? *

*Febe, the highly intelligent house fly, is bored buzzing around the kitchen so she hitches a lift when her Hoomins take a trip in their car. Being highly intelligent she can read the speedometer and convert the mph reading into m/s. She quickly becomes bored in the car and decides to fly directly from one side of the car to the other when the car is traveling in a straight line at 4 m/s. Being a mathematics prodigy, she realises that her speed relative to the ground is 5 m/s. How fast is she flying across the car? *

*A 1 ton car is heading due North at 60 mph. A 2 ton truck is heading due East at 30mph. There is sheet ice all over the intersection and the truck cannot stop. The truck smashes into the side of the car and the pair forms one tangled mess of bent metal. (Fortunately the drivers were wearing seatbelts and the air bags did their job. Nobody was injured.) In which direction does the mangled mess travel? (Hint: conserve momentum not energy). *

*A 1 ton car is heading due North at 60 mph. A 2 ton truck is heading due East at 30mph. There is sheet ice all over the intersection and the truck cannot stop. The truck smashes into the side of the car and the pair forms one tangled mess of bent metal. (Fortunately the drivers were wearing seatbelts and the air bags did their job. Nobody was injured.) At what speed does the mangled mess initially travel? (Hint: conserve momentum not energy). *

*For no clearly explained reason I am standing on a set of bathroom scales in a lift within a tall building. When the lift is stationary I weigh 41kg. When the lift moves I see my weight increase to 60.5kg. Describe the motion of the lift: *

*There is a right angled triangle with a hypotenuse of unit length. Denote an angle (other than the right angle) as alpha. Given that the side adjacent to the angle is of length cos(alpha) and the side opposite the angle is sin(alpha), evaluate the sum. *

* An American salesman flies over to London (UK) and for reasons best known to himself carelessly steps out of a first floor window. He breaks his leg. Why?*

* A elderly monk is arranging the annual charitable gift. He will put bank notes in two envelopes such that one envelope has twice the amount in the other. The number of notes will be undetectable within the heavy envelope. It is required that anyone who opens an envelope does not know if they have the high amount or the low amount. Given that the bank notes available to the monk only occur in units of $1, $2, $5, and $10, which statement is acceptable? Inspired by a comment from Jeff Jordan concerning Two Envelope Paradox*

* Jane sees the following text written on the blackboard in the classroom, evidently left over from a previous lesson. X = X + 1 Which statement is true?*

* In a singles tennis tournament of 64 players, the winner of each game goes forward to the next round. Two sisters are both excellent tennis players. Given that each sister wins all their matches until they meet each other, what is the probability that they meet each other at the final?*

* My black credit card has a 16-digit number. Credit cards have the digits 0-9 with equal probability in each digit position. What is the probability that the sum of the first fifteen digits is equal to the sum of the last fifteen digits?*

*In a fictitious far away country income tax is not charged on the first $10,000 of annual income. After that, tax is charged at a 20% rate on the amount beyond the $10,000 allowance. If income exceeds the allowance by more than $30,000, however, earnings beyond that amount are taxed at 40%. Arthur earns $20,000 per year. Barry earns $50,000 per year. How much more income tax (as a ratio) does Barry pay? *

*Jane wants to use her soapbox racer at the park where there is a sharp-edged ramp. Last time she tried it the bottom of the racer struck the corner of the ramp, making a horrible sound; this was extremely uncool. (Notice that the base of the racer always lines up with the axis of the wheels in these designs.) What should Jane do to restore her cool reputation? *

*In factories where food items are packaged, one clever technique for optimally filling bags is to fill 12 nominally equal hoppers with the food, then computer select the 4 which give the closest fit to the required weight. This is better than taking food items such as crisps and putting tiny broken pieces in to make up the required weight. It is also cheaper for the manufacturer to not greatly exceed the minimum weight, and effectively get paid less for each gram of food as a result. How many combinations does the computer have to check to get the optimal selection?
*

*John got back to his car too late and now he is locked into an outdoor car park for the night. There is an escape path, but it involves driving down a fairly steep grassy slope. He has correctly drawn a diagram of the problem, but can’t quite finish it off. The axle of the wheels is roughly in line with the underside of the car. Hitting the underside of the car on the corner of the slope will mean a tow truck will be needed and there may be costly damage done as well. The faint line above the ground is the path that the car axle travels as it rolls along the ground. This is called *

*Mary has a cat, and a rat, and a hat, and a mat. If the cat is on the mat, and rat is in the hat, but the hat is on the mat, where is the rat?*

*There are 100 unique numbered components in a bin. You select 5 components at random. You then sort the components into numerical order. How may different selections can you make (you restore the components to the bin after each selection)? *

*A motorboat is in the middle of a fast flowing river heading directly for some rocks that also happen to be in the middle of the river. Its engine failed 5 minutes ago so the boat is just being pulled along by the flow. Seeing the rocks in plenty of time the skipper pushes the rudder hard over to the left. The river channel is safe either side of the rocks. What happens next? *

*Three ropes join at a ring which is free to move relative to the ground. The forces and directions are shown at one instant. The sketch is not necessarily drawn to scale. What happens next? *

*It is a serious mistake to think that computers can solve any numerical problem almost instantaneously. Whilst addition and subtraction are very fast, multiplication can take twice as long as that, and division can take ten times as long as addition. In this example the values subscripted by N are to be calculated hundreds of thousands of times. The unsubscripted values are constants. Rearrange the equation to minimise the computational time. We are after the form of the equation; the values of the constants vary as necessary to make the result correct. *

*A hospital runs a queuing system for non-urgent surgical procedures. This queue always has 100 patients in it and one patient is operated on each day (7 days a week). If a patient is not available on the day of surgery they lose their place to a brand new patient who would otherwise have gone elsewhere. (This is easier than rescheduling 100 appointments.) Roughly 5% of patients are not available or do not show up for their procedures. What is the average waiting time reported to the managers of the hospital? *