My Favorite Math Puzzles

by John P. Pratt
last updated 4 Sep 2015

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None of the following puzzles has a trick answer, or unwarranted assumptions. They are arranged approximately in order of difficulty; the first few can be done by elementary school children, but they may need to be taught how to think. I tried to include a wide variety of types of puzzles, so that each would teach a new lesson in finding creative solutions by breaking a variety of kinds of mental blocks. Moreover, most of them can be solved mentally by using a judicious point of view.

1. Three men each paid $10 to share a $30 hotel room. Later, the manager felt bad about overcharging them for a $25 dollar room, so he gave the bellboy 5 one-dollar bills to return to them. The bellboy returned one dollar to each of the men, but kept two for himself. So the men paid $27 for the room, plus $2 to the bellboy for a total of $29. Where did the extra dollar go?
2. If you had fifty common U.S. coins that added up to exactly a dollar, how many dimes would you have?
3. A race car is going around a track that is a square, one mile on a side. If it goes 60 mph around the first two sides and 30 mph on the the third side, how fast must it go to average 60 mph all the way around?
4. If a hiker averages 2 miles/hr uphill and 6 miles/hr coming back the same trail, what is his average speed going both ways?
5. A poll taker was fired after reporting that she interviewed 100 people of whom 78 drink coffee, 71 drink tea, 48 drink both. Why is her report impossible?
6. Tom and Betty have the same birthday and are both in their twenties. He is four times as old as she was when he was three times as old as she was when he was twice as old as she was. How old are they?
7. The coach, who also taught math, said he'd give each of the eleven boys on the football team the same amount of money to buy candy, provided that each spent the money differently on the 2, 3, 4, or 6 cent candy. If he gave each the smallest amount with which that could be done, how much did each receive?
8. If 6 anteaters can eat 6 ants in 6 minutes, how many anteaters would it take to eat 100 ants in 100 minutes? Another variation is: If a chicken and 3/4 can lay an egg and 3/4 in a day and 3/4, how many chickens does it take to lay a dozen eggs lay in a week? (Jr. High).
9. Four black cows and three brown cows give as much milk in five days as three black cows and five brown cows give in four days. Which color cow gives the most milk?
10. How many times a day do the hour and minute hands on a clock line up exactly with each other?
11. Without writing down any trial solutions, prove that the following digit substitution problem has no solution:

               E L E V E N
               - T H R E E
                 E I G H T
    

12. A tournament had A and B divisions, each with from 3 to 9 contestants. Each division was a round-robin, in which every contestant plays every other contestant. One contestant was not allowed to register late because there were already so many matches to be played. However, that decision was reversed when he showed that allowing him to enter would actually decrease the total number of matches, provided that his friend were allowed to change divisions. How many contestants ended up in each division?
13. Two professors met again after several years and one said he was now married with three children. When the other asked their ages, the first said that the product of their ages is 36. The second asked for another clue and the first asked if he could see the number on the house across the street. When the other said yes, the first said that the sum of their ages equaled that number. The second said he still could not determine their ages. Then the first said that his oldest child has red hair. Finally the second knew their ages. What were their ages and how did he know?
14. A girl spent exactly one dollar (no sales tax) and bought exactly 100 pieces of candy, including some at $ .05, $.02, and 10 for $.01. How many pieces of each kind of candy did she buy?
15. Broke Bill enters the bank to cash a check. The teller gets confused and where cents is written gives him paper dollars, and where dollars are written, gives him cents. As Bill gathers the money, he unwittingly drops a coin on the floor, losing it. When Bill gets home and counts the money, he discovers he has exactly twice the amount that was written on the check. What was amount on the check?

    Advanced Variation: If he had dropped and lost two coins of different denominations, then what would the amount on the check have been?

16. On a TV quiz show a contestant is shown three closed doors and told that two of them have nothing behind them, but that one has a new car as a prize behind it. The contestant makes her choice of doors where she thinks the prize is. Then one of the other two doors is opened where there is no prize, and the contestant is asked if she would like to change her guess. Do the odds favor changing the guess? Why or why not?
17. If Tom is twice as old as Howard will be when Jack is a old as Tom is now, who is the oldest, next oldest and youngest?
18. A spoonful is removed from a cup of wine and placed in a cup of water and stirred well. Then a spoonful is removed from the resulting solution and put back into the cup of wine. Is there more water in the wine or vice versa?
19. A spider is one foot from the floor in the middle of 12 foot square end wall of a 30-foot long room. A fly is situated in the middle of the other end wall, 1 foot from the ceiling. What is the shortest walking distance from the spider to the fly?
20. A book has 600 pages and an average of 1 typographical error per page. What is the probability of finding n of them on any one page (for n = 0, 1, 2, or any integer). (Hint: use a Poisson distribution.)