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?
Jer has the correct answer. (The answer might be rephrased to say: "
the 15 possible sums that include A are..." )
And yes, sorry, I had a typo in the third example. I intended to write 28, not 26. I had a posted long list
here of sets and sums (including 1113)..... which solve the riddle. I will release the solution, but first ask as a bonus question:
What is the general formula for the coefficient Coef(n,k) for the sum of sums for n numbers: N_i, on k cards, (n taken k at a time with replacement, summed), as in:
sum(sums) = Coef(n,k) {sum i=1,n} N_i ?
e.g, Coef(5,3)=21
Edited on December 31, 2022, 1:42 am
bonus
