In the puzzle
Magic Trick
we are asked to find five numbers that a magician
has put on each of three cards such that adding any three of these numbers
yields a unique sum. The sum is always unique in the sense that it allows
the magician to say which three numbers were chosen.
In a simpler version
of the problem, the same five numbers have been written on each card, so, an individual number may be added more than once. Each sum is unique, and we are asked to find
the numbers that allow the trick to work and also give the minimum sum of all
possible sums.
The answer is (1, 2, 5, 16, 25) with a sum of all sums of 1029.
The next closest answer is (1, 2, 5, 17, 27) with a sum of sums of 1092,
and the next is 1, 3, 6, 15, 26 with a sum of 1113.
Another non-optimal
answer is (3, 6, 7, 16, 31) with a sum of sums of 1323.
It is noticed that
all answers to this optimization problem are different by multiples of 21. Why is this?
One might observe the sum of all sums is always 21 times the sum of the 5 numbers.
Call the numbers A,B,C,D,E the 15 possible sums are
A+A+A
A+A+B
A+A+C
A+A+D
A+A+E
A+B+B
A+B+C
A+B+D
A+B+E
A+C+C
A+C+D
A+C+E
A+D+D
A+D+E
A+E+E
and the sum of these is 21(A+B+C+D+E)
Actually there appears to be an error in this problem. The third solution in your post on the original problem lists 1,3,6,15,28
which does have sum 53=1113/21. You gave the last number as 26.
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Posted by Jer
on 2022-12-30 14:32:42 |
Solution
