## 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* + 10*d* + 25*q* = 00, you get
9*d* + 24*q* = 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