# Some simple and not so simple maths problems

These come from different sources. Some come from books and are a few hundred years old. I learned others from my friends. One is especially memorable. It was posed (and solved) on a rainy night in Australia, as we drove west from Sydney, trying to find a hole in the clouds to see the Leonids meteor shower....

1. A cask of kvas. A man drinks a cask of kvas (ancient Russian soft drink) in 14 days. Together with his wife they drink a similar cask in 10 days. How many days does it take the wife to drink it up by herself?

2. The twelve people. Twelve people carry 12 loaves of bread. Each man carries two loaves, each woman has a half, and each child holds a quarter. How many men, women and children are there?

3. A fake coin. Among nine coins there is a fake one which is lighter. How to find it using a balance if only two weighings are allowed?

4. Dangerous crossing. Four creatures A, B, C and D come to a river at night. The bridge is very thin and narrow, and can only hold any two of them at a time. Besides, it is dark and they need to keep their torch on while on the bridge. It takes A one minute to cross the bridge, B - 2, C - 5, and D - 8 minutes. Can they all cross to the other side if the batteries in the torch last only 15 minutes?

5. The seven bridges. The old city of Konigsberg (now called Kaliningrad) had two islands and seven bridges.

Its citizens always wanted to know if one could traverse all the bridges without taking the same bridge twice. What do you think?

6. 64-2. Two diagonally opposite squares have been cut away from a chess board. Is it possible to pave it with domino pieces? (Each domino piece covers two adjacent squares on the board.)

7. Merchant's problem. A merchant had a 40 pound weight. Once it fell and broke into four pieces. The merchant was very sad, but then realised that with the four new ``weights'' he could weigh any amount with an integer weight between 1 and 40 pounds by his balance. What were the weights of the pieces? (Remember that when weighing something you can put weights in both pans.)

8. A tricky catch. Three men on a fishing trip stopped by a river and fished untill darkness. They put their catch in a bucket, had a quick meal and went to sleep. When one of the fishermen woke up early next morning the other two were still asleep. He counted the fish in the bucket, realised that the number could not be divided by three, threw one fish back into the river, took one third of what was left, and left quietly. When the second man woke up he did not notice that the first one was already gone. So he counted the fish, saw that the number was not divisible by three, threw one fish into the river and left with his "third". Believe it or not, the same thing happened to the third fisherman. Upon waking up, he did not notice that the other two had already left, so he counted the fish, threw one fish into the river, took one third of the remaining fish and went home. How many fish were originally in the bucket? (Find the smallest possible number.)

9. Ship spotting. Every day at noon (GMT) an ocean liner of a certain shipping line leaves Southampton for New York, and another one leaves New York for Southampton. It takes the ships exactly seven days to complete the journey. How many liners bound for Southampton can a passenger spot during a sailing to New York?

10. A pizza and a half. If a student and a half can eat a pizza and a half in an hour and a half, how many pizzas will 12 students eat in 3 hours?

11. The hungry mouse. A mouse will gain 2 grams of weight every day eating as much cheese as possible. Not eating results in losing 3 grams of weight per day. If the mouse gained 5 grams over 20 days, how many days were spent not eating for the mouse?

12. Fixing the chain. Suppose you have 5 sections of the same type chain, each of which contains 3 links. You want to put them all together to make one length of chain. It costs £2 pounds to cut a link and £4 to weld a link. What is the most inexpensive way to make the new 15 link chain?

13. Lots of lockers. A school has a hall with 1,000 lockers numbered 1 to 1000, all of which are closed. A thousand students start down the hall. The first student opens every locker. The second student closes all the lockers whose numbers are multiples of 2. The third student changes (i.e., closes an open locker or opens a closed one) for all lockers with numebrs that are multiples of 3. The 4th student changes all multiples of 4, and so on. After all students have finished with the lockers, how many lockers are open and which ones?

14. A pilgrimage. A monk from a monastery at the foot of a mountain, begins to ascend it at sunrise. He follows a track, takes rests to catch a breath or to have a meal, and arrives at the summit at sunset. After spending the night there, he begins his descent at sunrise and follows the same track downhill, with occasional stops, to arrive back at the monastery at sunset. Must there be a spot along the track which the monk visited at the same time on both days?

15. A wolf, a goat and a cabbage. You need to ferry a wolf, a goat and a cabbage in your care, to the other side of a river. You have a boat which can only hold you and one other item (either the wolf, or the goat, or the cabbage). How can you fulfil your task if the wolf will eat the goat, if left together unsupervised, and so will do the goat to the cabbage!

16. A New York story. A young man lives in Manhattan near a subway station. He has two girl friends, one in Brooklyn, one in The Bronx. To visit the girl in Brooklyn he takes a train on one side of the platform; to visit the girl in The Bronx he takes a train on the other side of the same platform. Since he likes both girls equally well, he simply takes the first train that comes along. The young man reaches the subway platform at a random moment each Saturday afternoon. Brooklyn and Bronx trains run every 10 minutes. For some reason he finds himself spending most of his time with the girl in Brooklyn: in fact on the average he goes there nine times out of ten. Why does this happen?

17. Safe keeping. An international committee consists of 9 people. The committee keeps important papers in a safe. How many locks must the safe have, and how many keys for them should be produced, so that it can be opened precisely when at least 6 members of the committee are present?

18. Dropping the ball. Someone is given two identical balls that can break when dropped from a certain unknown height. To find what this height is, the person is allowed to drop them from any floor of a 100-story skyscraper. What should be the strategy that would allow one to find at what floor the balls break using the smallest number of drops?

19. Hostage drama. A group of 100 people have been taken hostage. Their captors explained that they are going to put a white or black hat on the head of each hostage, in such a way that they would not know its colour. The hostages will then be formed into a single file, so that each can see all other hostages in front, but not those behind. Each hostage will then be asked to call the colour of his or her hat, starting from the back of the file. Those who get the colour correctly are set free. Those who get it wrong will be shot.

The hostages will not be allowed to communicate once the hats are put on. However, they have a few minutes now to work out a strategy that would allow as many of them as possible to be saved. What should this stragy be and how many can be guaranteed to be freed?

20. Prison drama. A certain mumber of people are put in a jail. Before they are taken to their solitary cells, their jailers explain that each day one of them, chosen randomly, will be taken into a room that contains a two-position switch, and will be alowed to change its position if they want to do so. The prisoners are allowed to nominate one of them as their leader. The leader (who will also be taken to the switch room, on equal footing with the rest) is allowed to declare at some point that all prisoners have been to the switch room. If he is right, the prisoners will be set free. If he is wrong, they will remain in jail forever.

Before they are separated, the prisoners can discuss the matter and choose their leader and the strategy that may allow them to go free one day. What should this be?

21. Unlikely escape. A man is in the middle of a circular field surrounded by a fence. There is also a dog that can run along the fence. The maximum speed of the dog is four times that of the man. Will the man be able to escape, i.e., reach a point in the fence before the dog gets there?

22. What time sunrise? Vladimir Arnol'd (1937-2010) was one of the greatest 20th century Russian mathematicians. He told the following story:

“The first real mathematical experience I had was when our schoolteacher I. V. Morozkin gave us the following problem: Two old women started at sunrise and each walked at a constant velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what time was the sunrise on this day?

I spent a whole day thinking on this oldie, and the solution (based on what is now called scaling arguments, dimensional analysis, or toric variety theory, depending on your taste) came as a revelation. The feeling of discovery that I had then (1949) was exactly the same as in all the subsequent much more serious problems--be it the discovery of the relation between algebraic geometry of real plane curves and four-dimensional topology (1970) or between singularities of caustics and of wave fronts and simple Lie algebra and Coxeter groups (1972). It is the greed to experience such a wonderful feeling more and more times that was, and still is, my main motivation in mathematics.”

23. Crossing ladders. A 20 foot ladder is placed in a straight alleyway from the lower left corner and leaning on the right wall. One ladder-width further down the alleyway, a 30 foot ladder is placed from the lower right corner and leaning on the left wall. The point where the two ladders cross is 10 feet above the ground. How wide is the alleyway?

Can you solve this problem?

Do not rush to use these hints!

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Dr. Gleb Gribakin / phone: +44 (0)28 90971936
E-mail: g.gribakin@qub.ac.uk