Solutions to Claude Berge's puzzles

Puzzle Solutions

Puzzle 1

Blue plays at b3. This piece is connected to the left border by a3/a4 and to the long blue chain by c2/b4. The long blue chain is connected to the right border by e1/d5.

Puzzle 2

This is identical to Peit Hein's Puzzle 7. You may wish to try the other puzzles on that page before consulting the solutions.

Puzzle 3

Red threatens to play at 2, 4 and 6.

Puzzle 4

The simplest solution is perhaps the sequence of forcing moves by Red below, although 1 or 3 at 5 may be quicker. The solutions to puzzles 3 and 4 are examples of climbing.

Puzzle 5

This puzzle is usually presented as "Blue to play and win", but this is too easy, since Blue can win by playing at k3, leaving the bottom part of the puzzle irrelevant. But this is likely not what Berge had in mind, since in his manuscript "L'Art subtil du Hex", he instead outlines a solution to the bottom part of the puzzle. Berge's intention is a bit difficult to discern, since different drafts of his manuscript contain slightly different versions of this puzzle, and a final definitive version was never published. He also refers to this as a "study" rather than a puzzle. Regardless of what Berge's original intention was, since it is in fact a second move win for Blue, "Red to move and Blue to win" seems like a good interpretation of it.

The puzzle decomposes into two disjoint parts. In order to win, Red would need to connect k4 to the top edge while also connecting l7 to the large red group that is already connected to the bottom edge. However, Red cannot achieve both of these goals simultaneously. If Red plays in the bottom part of the board, Blue can play k3. This leads to a 4th row ladder that Red cannot win. For example:

On the other hand, if Red plays in the top part of the board, Blue can play k8.

With this move, Blue threatens to immediately connect right via Edge template IIIa. A typical sequence of follow-up moves is:

Now the blue stone at k11 is safely connected to the right via Edge template IV2a, and is also connected left via the double threat at the hexes marked *.

See also

• Back to Claude Berge's puzzles
• Main page: Puzzles