Using Your Marbles


Your friend Carol loves to prepare gifts of marbles in bags for children. On Monday, she has 5 red marbles, 6 blue marbles, and 7 white ones.

Warm-Up

How many different bags can she pack so each bag will have a different collection of marbles? Two bags are different if for at least one color, one bag has a different number of marbles of that color from the other. For example, a bag with two red marbles and one blue is different from a bag having one red and one blue or a bag having one red and two blue.

Solution to Warm-Up

For brevity, the colors will be represented by their first letters: R (red), B (blue), and W (white). Carol can pack ten different bags: R, B, W, RR, BB, WW, RB, BW, RW, BWW.

That was just the warm-up.

  1. On Tuesday, she again starts with 5 red marbles, 6 blue marbles, and 7 white ones. In order to make the bags more varied, she wants to pack the bags so that each bag differs by at least two “in-or-outs” from any other bag. An in-or-out consists of inserting a marble into a bag or removing a marble from a bag. So, for example, R and RR differ by only one in-or-out (insert R). On the other hand, RW and RB differ by two (remove W and insert B). How many different bags of marbles can Carol create that differ by at least two in-or-outs?

On Wednesday, Carol is given a bag that she cannot open.

  1. She knows she has 18 marbles. She also knows that there is at least one red, at least one blue, and at least one white. Knowing only this information, but before seeing the marbles, Carol receives a phone call. “Can you guarantee to be able to give each child a bag with a different collection of marbles (i.e., at least one in-or-out apart), if there are eight children at a party? If not, then can you make this guarantee if there are seven children? If so, then how about nine?” How should Carol answer?

  2. On Thursday, she again knows only that she has 18 marbles and at least one of each color, and is in the same situation as Tuesday as far as wanting the children to enjoy variety. So she wants each gift to differ from every other by at least two in-or-outs. How many children can she guarantee to prepare bags for in that case?

  3. On Friday, her friend Diane packs the bags. Diane assures Carol that (i) every bag contains a different (some difference of number in at least one color) collection of marbles, (ii) that there are 18 marbles to start with, (iii) there are more white than blue and more blue than red, and (iv) that the maximum different ones she could pack was seven. Assuming that Diane is an extremely capable packer, what is the maximum number of red marbles there could have been at the beginning?




Puzzles for Programmers and Pros
Puzzles for Programmers and Pros
ISBN: 0470121688
EAN: 2147483647
Year: 2007
Pages: 81
Authors: Dennis Shasha

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