1. Making Nine
[From Mathematical Bafflers, extended by Sam Rebelsky.]
Arrange the digits 0 to 9 in fractional form so that xxxxx/xxxxx = 9. Each digit should appear exactly once.
Alternately, assign each digit from 0 to 9 to a letter from A to J so that ABCDE/FGHIJ = 9.
2. Chessboard Squares
[From Thinking Mathematically, p. 19]
It was once claimed that there are 204 squares on an ordinary chessboard. Can you justify this claim?
3. Ins and Outs
[From Thinking Mathematically, p. 175]
Take a strip of paper and fold it in half several times in the same fashion as in the Paper Strip problem. That is, always place the right end on top of the left. Unfold it, keeping the left end in place, and observe that some of the creases are IN and some are OUT. For example, three folds produce the sequence
in, in, out, in, in, out, out
What sequence would arise from 10 folds (if that many were possible)? Explain the patterns.
4. Dividing by Eleven
[From Thinking Mathematically, p. 167. Modified by Sam Rebelsky.]
To check whether a number is divisible by 11, sum the digits in the odd positions counting from the left (the first, third, ...) and then sum the remaining digits. If the difference between the two sums is divisible by 11, then so is the original number. Otherwise it is not.
5. Dividing by Three
It is a "well known" fact that a number is divisible by three only when the sum of the digits in the number is divisible by three. Is it true? Can you figure out why? Is it true for any other number?
6. Sharing Gumballs
[From the notes of Emily Moore (who notes the problem comes from p. 4 of "MG"). Modified by Sam Rebelsky.]
The Petersons wheel their twins past a gumball machine. The twins shout that they each want a gum ball. From past experience, the Petersons know that if they give the twins different color gumballs, unspeakable horrors will follow.
The Petersons look closely at the machine and observe that there are 6 red balls, 4 blue balls, 5 white balls, and 7 green balls. How many gumballs must they buy to ensure that they can give both twins the same color gumball?
The Tennesons have a more difficult problem. They have triplets and still need to ensure that each child has the same color gumball. If they get to the machine before the Petersons, how many gumballs will they have to buy to ensure that they have three of the same color?
The Tinkers have a related problem. They also have triplets, but the triplets object to having everything the same. How many gumballs must the Tinkers buy to ensure that they have three different colors?
The McCaughey's have an even worse problem, given that they have septuplets. If they get to the machine first, how many must they buy? What if there are 20 balls of each of four colors?
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