Puzzles to Play With

Transcription

1 Puzzles to Play With Attached are some puzzles to occupy your mind. They are not arranged in order of difficulty. Some at the back are easier than some at the front. If you think you have a solution but are not sure, you can send it to me at richardson@math.wichita.edu and I will check it for you. Please enter Math Circle in the subject line of your .

2 Puzzles 1. Train. A train one mile long travels at the rate of one mile a minute through a tunnel which is one mile long. How long will it take the train to pass completely through the tunnel? 2. Count the Bricks. One side of a chimney on a flat roof looks like the figure on the right. No bricks are cut. The small squares represent the ends of bricks. How many bricks are used in making the entire chimney? 3. Next Number. The following sequence of numbers has two possible replacements for the question mark. Can you find them both? 17, 19, 23, 29,? 4. The Chemist A chemist had been working on discovering a liquid in which all substances will dissolve. One day he announced he was able to make a full bottle of this liquid. What s wrong with his claim? 5. How Many Quarters? Place a quarter on a flat surface. Place another quarter to the right and just touching the first one. Keep going around the circle (see figure below), with each quarter touching the center one and the one just placed. By the time you place the last possible quarter, how many quarters will you have placed around the center one? 6. A Number Pyramid Suppose the numerical pyramid below kept going. Without writing down the following rows, what would the sum of the numbers in the tenth row be?

3 7. Secret Code. Joey sent his friend Kevin a secret coded message using the standard code A = 1, B = 2,..., and so on. However in one coded word he forgot to leave space between the numbers. What he sent was Can you find what this word is? (Actually there are two possibilities.) 8. The Bleachers. When the bleachers at a high school basketball game were full they held 559 people. Assuming each row held the same number of people, how many rows were there? 9. Divisibility. What is the smallest base ten number containing only 0 s and 1 s that is divisible by 6? 10. Buying a Hat and Coat. If a hat and coat costs $110 and the coat costs $100 more than the hat, how much does the coat cost? 11. Back Where You Started. Take a two-digit number and square it getting a three-digit number with distinct digits. Reverse the three distinct digits of this square. Take its square root. Reverse the two digits of that number, and you have the original two-digit number. What is that number? 12. Three Player Game. A game is played by 3 players in which the one who loses must double the amount of money that each of the other 2 players has at the time. Each of the players lose one game at at the conclusion of three games each man has $16. How much money did each man start with? 13. Tree Stump. An evergreen tree is in the shape of an equilateral triangle. As it happens, the shape of the tree makes it possible to draw a circle around the tree as shown in the figure below. How tall is the stump of the tree in relation to the entire tree?

4 14. Calendar Dates. The date July 14, 1998 has the interesting property that if you write the date in the numerical form 7/14/98, it has the property that the month multiplied by the day is equal to the year. During the 1950s, which years had no dates of this form? 15. Calendar Dates Again. A set of three whole numbers {a, b, c} form what we call a Pythagorean triple if a 2 + b 2 = c 2. This is a property of right triangles given in the Pythagorean Theorem; that is, the sum of the squares of the two legs of a right triangle is equal to the square of the hypotenuse. The date March 4, 2005, written in numerical form 3/04/05 is a Pythagorean triple, since = 5 2. Can you find some other Pythagorean triples in the 21st Century? 16. The Striking Clock. If it takes 7 seconds for a clock to strike 7, how long does it take to strike 10? 17. Family Matters. 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? 18. Same Sum. Arrange the numbers 1 through 9 in the square below so that each of the three columns has the same sum. Now if you can arrange the numbers such that all the rows and all the columns have the same sum, you will have a magic square. Can you do that?

5 19. Follow the Dots. 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. 20. The Gold Chain. 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? 21. Measuring. A container holds exactly 8 quarts of liquid and is filled to the top. Two empty containers are also available, one of which holds 5 quarts and the other 3 quarts. Without the use of any other measuring devices, the liquid can be divided into equal parts of 4 quarts each simply by pouring from one container to another. How is this done? 22. Rabbits and Chickens. A man was asked how many rabbits and chickens he had in his yard. He replied, Between the two there are 60 eyes and 86 feet. Although this reply was not exactly responsive, cn you determine how many rabbits and chickens the man had? 23. A Box of Marbles. There are 4 black, 8 red and 5 white marbles in a box. How many marbles must one take out (without looking at them) to be sure that there are at least 2 of one color among the marbles selected?

6 24. Making Money? A man bought something for $60 and sold it for $70. Then he bought it back for $80 and resold it for $90. How much profit, if any, did he make?