Poolball Puzzle

A set of 21 poolballs has been arranged in four rows, packed triangularly, with a couple of gaps. The balls are numbered from 1 to 21. The puzzle is to figure out where each ball is.

You are shown to positions of the balls. But instead of showing the number on each ball, instead you are shown the sum of the numbers on any balls that touch that ball (and including that ball). You are also shown the sum of the numbers of the balls in each row.

Here they are:

37 43 51 53 27 <

SPOILER

1 17 9 6 15 3 16 xx 19 4 2 11 5 10 7 13 8 12 xx 18 20 21 14 xx

Please reply to ilan at cedara dot com

You beat me to it! I noticed that you could get 25 equations in 21 unknowns which seemed likely to have a unique solution. So I wrote up a little gaussian elimination program (with column pivoting), entered the equations in a matrix, ran the program, and out popped the solution. Not exactly an inspired method but it worked.

Yep, it's an over-specified system of equations; gaussian elimination is one way to solve it.

There is another way to solve this type of puzzle that I find more enjoyable. It's similar to the graphical method you posted for your magic hexagon solution. I've appended it below.

ObPuzzle: can this puzzle be solved without using the row sums?

Bob H

Yes! You do not need the row sums nor the fact that the numbers include all the integers from 1 through 21. The other 21 equations are linearly

and Glenn C. Rhoads replied:

Cool. I'll have to do the same experiment, and then see what the equivalent graphical equations are.