There are in reality three terminations possible to the problem—the single ball, the couple, and the tierce; that is, you may have only one left, or two placed diagonally, such as 9-17, 25-29, or a system of three in a straight line, 9-16-23. By the “equivalents” you can always succeed in solving the problem desired.
We will now point out four transformations which are very easy to effect, and result from the rule of “equivalents.”
1. Replacement of the two balls, situated on the same line and separated by an empty cup, by one put into that cup. Thus I can replace 23 and 25 by a single ball at 24.
2. Suppression of tierces. And by the above movement I suppress the tierce 9-16-23.
3. Correspondent “cases” are two holes situated in the same line and separated by two cups. If two corresponding cups are filled, I can suppress the balls which occupy them. So I can put aside 4 and 23.
Fig. 868.—Correspondents and equivalents.
4. It is permissible to move a ball into one of the correspondent cups if it be vacant; thus I can put 10 into 29.
These are the four transformations which can be made evident with the rings, without displacing the balls. To do this we need have only seven rings large enough to pass over the balls and to surround the holes in which they rest. Let us take an example.
Solitaire with 33 holes (fig. 869). Final solution of the single ball.