X-Chain
One digit, an alternating chain that both starts and ends on a strong link — so one of its two ends holds the digit.
How it works
- 1Work with a single digit. A STRONG link joins two cells in a house where that digit has only those two homes left: if one is not the digit, the other must be. A WEAK link joins any two homes sharing a house: they cannot both be it.
- 2Build a chain that alternates strong, weak, strong, … and both BEGINS and ENDS on a strong link.
- 3Walk it: suppose the first cell is not the digit. The opening strong link forces the second cell on, the weak link forces the third off, and the alternation carries all the way to a closing strong link that forces the last cell ON.
- 4So whichever way the first cell turns out, at least one END of the chain holds the digit. Any cell seeing BOTH ends therefore cannot.
- 5The ends matter more than the length. A chain ending on a weak link proves nothing — there, both ends can be off at once.