»  Solutions to puzzles in my National Review Online Diary

  October 2006

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In my October diary I posed the following brain-teaser.

Sally has 100 US coins. They add up to $5.00. None of the coins is a nickel. What coins does she have?

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Solution

A complete solution is given below. The five numbers in each line refer to, respectively, cents, dimes, quarters, 50-cent pieces, and dollar pieces. The last two are legal tender, so I don't know why they should be excluded.

Notice, however, that none of the 19 solutions ends with "-0-0." This means that there is no solution in only cents, dimes, and quarters; so if you do exclude half-dollar and dollar coins, then there is no solution.

(The reason there is no cents-dimes-quarters solution is that if you subtract one of the governing equations, c + d + q = 100, from the other, c + 10d + 25q = 00, you get 9d + 24q = 400. Since, for any whole numbers d and q, the left-hand side of that divides exactly by 3 but the right-hand-side doesn't, there can be no whole-number solution.)

Every one of these solutions adds up to five dollars, and every one uses a hundred coins.

60-39--0--1--0
65-31--3--1--0
70-23--6--1--0
75-15--9--1--0
80--7-12--1--0
75-20--1--4--0
80-12--4--4--0
85--4--7--4--0
90--1--2--7--0
70-28--0--1--1
75-20--3--1--1
80-12--6--1--1
85--4--9--1--1
85--9--1--4--1
90--1--4--4--1
80-17--0--1--2
85--9--3--1--2
90--1--6--1--2
90--6--0--1--3