Answers 355- AT THE BROOK
A B
15 16 15 16
0 16* 15 5* *15 0 0 11
15 1* 0 5 0 15 15 11
0 I 5 0 15 15 *10 16
0 5 16 *14 16 10 0
16 15 6* 14 0 0 1015 2 (^0 6 0 14 15 10)
(^0 2 6 0 15 14) 9 16
2 0 6 16 13 16 9 0
2 16 15 7 13 0 0 9
15 3 0 7 0 13 15 9
0 3 7 0 15 13 8 16
3 0 7 16 12 16
3 16 15 8 12 0
15 4 (^0 8 0 12)
0 4 8 0 15 12
4 0 8 16 11 16
4 16 11 0
Every line shows a transaction. Thus, in column A, we first fill the 16
measure; then fill the 15 from the 16, leaving I, if we want it; then empty the
15; then transfer the I from 16 to 15; and so on. The asterisks show how to
measure successively I, 2, 3, 4, etc. Or we can start, as in column B, by first
filling the 15 and so measure in turn, 14, 13, 12, II, etc. If we continue A we
get B read upwards, or vice versa. It will thus be seen that to measure from I
up to 7 inclusive in the fewest transactions we must use the method A, but to
get from 8 to 14 we must use method B. To measure 8 in the A direction will
take 30 transactions, but in the B manner only 28, which is the correct answer.
It is a surprising fact that with any two measures that are prime to each other
(that have no common divisor, like 15 and 16) we can measure any whole
number from I up to the largest measure. With measures 4 and 6 (each divis-
ible by 2) we can only measure 2, 4, and 6. With 3 and 9 we could only measure
3, 6, and 9. In our tables the quantities measured come in regular numerical