Below is a collection of problems that I have found interesting. Some of them have been used for our Math Awareness Internet Problem Competition.

There is no particular order of difficulty in the listing. I just put them on as I have time. Theyare in an ordered list for reference purposes.

1. How many brothers and sisters are there in a family in which each boy has as many sisters as brothers but each of the girls has twice as many brothers as sisters?

2. A secretary finds that she has an extraordinary social security number. Its nine digits contain all the numbes from 1 through 9. They also form a number with the following characteristics: when read from left to right, its first two digits form a number divisible by 2, its first three digits form a number divisible by 3, its first four digits form a number divisible by 4, and so on, until the complete number is divisible by 9.

   What is the secretary's social security number?

3. Nine dots are arranged in a square formation in 3 rows of 3. Draw 4 straight lines, the second beginning where the first ends, the third beginning where the second ends, and the fourth beginning where the third ends so that each dot is on at least one line.

                   *       *       *
   
   
                   *       *       *
   
   
                   *       *       *
   

4. Place ten coins in such a way that they lie in 5 straight lines and on each line there are 4 coins. (There is more than one solution.)

5. The law for the sale of alcoholic beverages in some states provides that beer cannot be sold after a certain hour, but permits a customer to consume, after the curfew, what has been served before the deadline.

   As the curfew approached, each of 2 men ordered sufficient beer to anticipate his probable requirements. The first man ordered and paid for 5 cans of beer and the second man ordered and paid for 3 cans of beer.

   As the curfew sounded, a mutual friend approached and, being thirsty too, suggested that he be permitted to share in the refreshments which had so providentially been provided. Permission having been granted, the 3 men shared equally in dispatching the beer.

   The third man thanked his friends and then put down 80 cents in payment for the beer he had consumed, which was to be divided equitably between the other 2 men.

   How should the money be divided to be fair?

6. An automobile has traveled 20,000 miles.

   If 5 tires were used equally in accumulating this mileage, how many miles of wear did each tire sustain?

7. A game is played by 3 players in which the one who loses must double the amount of money that each of the other two players has at that time.

   Each of the 3 players loses one game and at the conclusion of the 3 games each player has $16.

   How much money did each person start with?

8. A man had no money but he had a gold chain which contained 23 links. His landlord agreed to accept 1 link per day in payment for rent. The man, however, wanted to keep the chain as intact as possible because he expected to receive a sum of money with which he would buy back what he had given the landlord. Of course open links can be used as payment too and "change" can be made with links already given to the landlord.

   What is the smallest number of links which must be opened in order for the man to be able to pay his rent each day for 23 days?

9. Peter: Where did you get that ugly polynomial "f" with so many unknown coefficients? It looks rather troublesome.

   Paul: It's just a polynomial with integer coefficients, and one of the roots is precisely the age of my daughter, Mary.

   Peter: Anyhow it looks rather tedious. Let me compute f(7)... Alas, that gives 77 not 0.

   Paul: You don't know Mary's age exactly, but she is older than 7!

   Peter: Then let me try with the age of my son, John ...
   But that gives 85!

   Paul: Come on, Peter! Mary is older than your son.

   How old is John? How old is Mary?

10. Two old women started at sunrise and each walked at a constant velocity. One walked from A to B and the other from B to A. They met at noon and, continuing without stopping, arrived respectivelyat B at 4 P.M. and at A at 9 P.M. At what time was the sunrise on this day?

11. A clever man has constructed four cubes of different weights. With these marvelous weights anda balance scale he is able to determine the weight, within a quarter of a pound, of any object which weighs from a quarter of a pound to ten pounds. That is, he is able to weigh objects that weigh 1/4 lb, 1/2 lb, 3/4 lb, 1 lb, 1 1/4 lb, ... , 9 lb, 9 1/4 lb, 9 1/2 lb, 9 3/4 lb and 10 lbs. What are the weights of the four cubes?

12. Each week a charitable lady gave a certain amount of money to several needy people. One day she mentioned to this group that if there were five fewer people to give money to she could give each individual an additional two dollars. As hard as they tried, the group could not convince five of its members to drop out. Instead at the next meeting an additional four people showed up. As a result each person got one dollar less than the original amount.

    Assuming that the lady has the same amount of money each week to distribute, how much did she have, how many people were in the original group and how much did each get?

13. An artisan has just made a new kettle shaped as in the figure below. This flat-bottomed kettle is twelve inches deep and holds exactly twenty-five gallons of milk. Assuming that the kettle's brim has a diameter twice that of the bottom, what is the diameter of the brim (to the nearest inch)?

    

    (Shape is a frustrum of a cone.)

14. (This is one of my all-time favorite problems.) Find the smallest positive integer N such that the units digit is 6 and if the 6 is erased and placed in front of the remaining digits, the resulting number is equal to 4N.

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