Here are a selection of puzzles. Useful as quick time fillers, during registration periods and to make A level pupils think! Many of the puzzles don't have answers - it is creativity and intelligence in the method that is sought.

Intelligence Test

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Fermi Problems

1. How many trees are in Canada?
Think about the area of Canada, fractional area occupied by trees, and space between trees (tree density).

2. Central Line
How many underground cars operate on the Central Line during rush hour?
Trains run approx. every 2 minutes. Each train has 8 cars. It takes about 80 minutes for a train to travel the length of the line.

3. How many ping pong balls can fit on a double deck London bus?
Think about volume of bus and volume of ball. An interesting exension is to consider packing fraction of spheres. The most efficient packing allows the spheres to take up 74% of the available volume.

4. How many piano tuners are there in New York?
Think about the population of New York, the fraction of people who have pianos, how often they are tuned and how long it takes a piano tuner to tune one piano.

5. How many people can fit inside a Mini?
Consider volume of car and volume of a person. One way is to consider person weighs 60kg and is made up of water with density 1kg/l or 1000kg/m³. The world record for a new BMW Mini is 21.

6. If you flattened out all the mountains and valleys in Wales would it be bigger than England. Is this a reasonable statement?
If you know that the area of is Wales 20,779 km² and the area of England is 130,410 km² then the average slope angle of land in Wales can be calculated. This assumes, of course, that England is flat. In the same way that a cos theta factor can be used for straight lines (eg. a triangle) it can also be used for areas. So arccos(22779/130410) gives us that the angle would have to be 81°.

7. How many tyres are there in use in the UK?
The answer needs to consider the number of cars (with 4/5 tyres?) and the number of buses/trucks/vans.

8. How many molecules come off a car tire with each revolution?

9. How many frames are in a Walt Disney animated movie?

10. What is the mass of a fully loaded Boeing 747?

11. What is the length in miles of all the Motorways in the UK?

12. How many square kilometers of surface would it take to supply the UK with all its energy needs if solar energy could be converted with 1% efficiency? Allow for night time, cloud cover, etc. The solar constant
is 1.35 kW/m².

13. If all the oxygen atoms breathed by Enrico Fermi over his lifetime are now distributed uniformly through the atmosphere, how many of these atoms do you breathe in with each breath?

14. Pick a nearby tree. Estimate the number of leaves on the tree.

15. How many electrons are there on the earth?

16. What is the probability of winning a million pounds on Who Wants to be a Millionaire by guessing? Assume everyone guesses, including the audience and the phone-a-friend.

17. If the conversion of electrical energy to light energy is 75% efficient, how many photons are emitted per second by a 40 watt fluorescent tube?

18. How many molecules will evaporate each minute from a saucepan of water left to stand in the shade on a dry summer's day?

19. How many hairs are there on a dog?

20. How much do the brakes of a car heat up in stopping at a set of traffic lights?

21. How many dimples are there on a golfball?

22. What is the mass of all the tennis racquets in the world?

23. A 10 year old car is badly rusted. If it rusted at a uniform rate, estimate how many iron atoms each day combine with oxygen to form rust?

24. If there were a mole of butterflies lying evenly distributed ever the earth, how thick a layer would they produce?

25. How long does it take for the electrons in a TV tube to get from the electron gun to the screen?

26. How many pieces of popcorn does it take to fill a room?

27. How many oxygen molecules that pass in and out of the lungs of an adult human in one day?

28. Estimate the mass of lead deposited per square metre per year in London due to airborne lead emitted from cars, if each litre of petrol contains about 2 grams of lead.

29. It is a clear day and the sun is shining straight down on an out door swimming pool. How much will the water temperature change in 5 minutes? The solar constant is 1.35 kW/m².

30. A helium-filled balloon is 50 cm in diameter. What is its diameter when taken to the bottom of the Atlantic ocean?

31. Estimate the magnitude of the gravitational attraction between a man and a woman as they stand talking to each other.

32. What is the thickness of a piece of paper in wavelengths of visible light?

33. Consider the possibility of a large country such as China organizing a "geophysical weapon", by having all the inhabitants of China jump off chairs onto the ground at the same instant. Assuming that the resulting energy could all be focused to one point on the earth, how many kilograms of TNT would this weapon correspond to? 1kg TNT yields about 4MJ of energy.

34. How big does a seed on the ground have to be to justify a bird in flying off a tree branch to eat it?

35. How many bricks are there in Bristol?

36. How likely is the existence of an extraterrestrial civilization?

37. How much area would a 1000 Megawatt solar power generator need? The solar constant is 1.35 kW/m².

38. What is the heat output of a human?

39. How long would a laser have to stay focused on a missile to ignite the chemical explosives in the warhead?

40. How small can a 1 GB memory be?

41. At what distance is the magnetic field from high voltage transmission lines the same as the Earth's magnetic field?

42. Astrology claims that the position of the planets at the time of our birth influences our lives. Calculate the relative gravitational attraction on a newborn baby by Jupiter, the hospital building and the mother.

43. What is the kinetic energy of a drifting continent?

44. How large a moon can you jump off of?

45. How much energy is required to boil the Earth's oceans?

46. How many truck loads would it take to cart away Mt. Everest?

47. Assuming one Santa Claus visits all Christian children on Christmas, how fast would he have to travel?

48. The Catholic Church believes that at communion you receive the body and blood of Christ. Consider whether the Catholic Church are justified in this claim. Consider the number of atoms that once were part of Jesus, are in the world and are in a typical communion wafer…

49. Given that a pencil contains 1.3g of Carbon and that a human being is around 18% Carbon, how many pencils could be made using the Carbon from one person?

Other Puzzles

How many times do you have to fold a piece of paper in half to reach the moon?
Pupils will need to know the Earth-Moon distance is 384403 km and the thickness of paper can be assumed to be 0.1mm. This illustrates the magnitude of powers quite well since 2 to the power of n multiplied by the thickness of the paper equals the distance to the moon. Solving this gives that the paper only needs to be folded 42 times.

Deal or No Deal (sort of)
There are three boxes. One box has a prize. You must pick a box. I know which box has the prize. After you’ve picked I remove one box which does NOT contain the prize. Should you change your choice, keep it the same or does it not matter?
You should change. The probability that you were correct with your original choice was 1/3. After I remove a box, the probability that the box you’ve picked is correct is still 1/3, but the probability that the other box is correct is must now be 2/3.

Water Levels
There is a brick in a boat on a small pond. If the brick is thrown over the side into the pond, does the water level rise, fall or stay the same?
When in the boat the brick displaces its weight in water. When in the water the brick displaces only its volume in water. Therefore the water level in the pond will decrease (slightly) when the brick is thrown in.

There are three light bulbs in a room with no windows. Each is individually wired to one switch outside the room, far from the door. You are only allowed to enter the room once. How can you work out which switch operates bulb?
Answer: Turn switch A on, wait a few minutes. Turn switch A off and B on. Enter room, there will be a hot bulb (which is unlit), a lit bulb and a cold unlit bulb.

Lowest Unique Number
Pupils compete with class to have the lowest unique number written down in their books. Interesting to discuss strategy with more able groups.

Death Due to Hats

The characters are arranged as shown. There are always two black hats and two white hats. They can't turn round, move or communicate with eachother. One person needs to shout out with certainty what colour hat they have on. If they don’t they will all be shot. Can they survive, and if so how?
They can survive. Consider the options. Label the people ABCD from the left. If the person on the far right (D) can see that the two people in front of him (B and C) have the same colour hat, he must have on the opposite colour so can call out. If, however, he doesn't call out it must mean that he can see the two people in front of him (B and C) have different colour hats on. The person second from the right (C) then knows that if the person behind him (D) doesn't call out, he has the opposite colour hat on to the person in front of him (B).

Making 24

Using four number fives and any mathematical operators can you make the number 24?

This is easy: 5*5-(5/5) = 25-1 = 24

A harder extension: can you make 24 with just two fives?

Answer: 5!/5

Making 24 again

Using the numbers 8, 8, 3 and 3 in any order and just the 4 basic mathematical operators make the number 24.

Answer: 8/(3-(8/3))

Poison Wine

A king has 1000 bottles of wine. One (and only one) is known to be poisoned with slow acting poison (which always kills a person after one month). The king has 10 slaves who can each drink one whole glass of wine. The king decides how to pour each glass - using wine from one or more bottles. How can he discover which bottle is poisoned?

Answer: Use binary. Label all the bottles from 1 to 1000. Pour each bottle into its corresponding glass. For example bottle 16 = 2^4 should be poured into only glass 4. Bottle 528 = 2^9 + 2^4 should be poured into glasses 4 and 9. Thus each bottle has its own unique combination of glasses and thus slaves killed. So if bottle 72 is the poisoned one slaves 3 and 6 will die since 72 = 2^3+2^6 = 8+64.

© 2016 Dr Matthew French All rights reserved.