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The Archery Butt

Answer :

"It hath been said that the proof of a pudding is ever in the eating thereof, and by the teeth of Saint George I know no better way of showing how this placing of the figures may be done than by the doing of it. 
Therefore have I in suchwise written the numbers that they do add up to twenty and three in all the twelve lines of three that are upon the butt."

I think it well here to supplement the solution of De Fortibus with a few remarks of my own. 
The nineteen numbers may be so arranged that the lines will add up to any number we may choose to select from 22 to 38 inclusive, excepting 30. 
In some cases there are several different solutions, but in the case of 23 there are only two. 
I give one of these. 
To obtain the second solution exchange respectively 7, 10, 5, 8, 9, in the illustration, with 13, 4, 17, 2, 15. 
Also exchange 18 with 12, and the other numbers may remain unmoved. 
In every instance there must be an even number in the central place, and any such number from 2 to 18 may occur. 
Every solution has its complementary. 
Thus, if for every number in the accompanying drawing we substitute the difference between it and 20, we get the solution in the case of 37. 
Similarly, from the arrangement in the original drawing, we may at once obtain a solution for the case of 38.

Medieval Brain Teasers