Solutions to puzzles

Piet Hein

See Solutions to Piet Hein's puzzles

Claude Berge

See Solutions to Claude Berge's puzzles

Bert Enderton

Puzzle 1

The unique winning first move is Red b4. (Blue can easily defend the bridges at e3/e4 and g2/g3. Red d3 is defeated by blue b5, red b5 is defeated by blue d3, and all other red moves are defeated by blue b4).

Notice that the red stone at a6 is already connected south via the ladder escape at e5. Also, red threatens to connect north via a double threat at b3 and d3; each of these would connect to the northern edge via Template IIIa. The two templates overlap at d1, but blue d1 is defeated via the following sequence of forcing moves:

Puzzle 2

A winning move is Red e3. This connects to the top edge via Template IIIa, and to the bottom edge via Template J5. The two templates overlap at one point d3, but if Blue plays there, Red can reply f3 to seal the connection up and down.

Puzzle 3

The unique winning first move is Red c3! (e2 is defeated by e3, d3 is defeated by e1, e3 is defeated by e2, and b4 is defeated by d3).

The following seem like horizontal's (Blue's) best tries from the above position.

1. d2	e2	2. d5	c5	3. d4	b3	4. c4	a5
1. d3	b3	2. b5	e3
1. e1	d2	2. d1	d1	3. c4	b3	4. e3	a5	or	4. b5	e3
........	....	2. c4	b3	3. c5	e3	4. e4	f3	5. e6	d5
1. d5	b3	2. d2	b2	3. c4	a5	4. a6	c5	5. b5	e3	6. d5	f5
........	....	........	....	3. b5	c5	4. c4	e3
........	....	........	....	3. b6	a6	4. b4	c5	5. c4	e3	6. e6	d4
1. c4	e3	2. e2	b3	3. d3	a5
1. b5	d4	2. d3	f2	3. f1	d2	4. c4	e2	5. e4	e3	6. c5	e5
........	....	2. e1	d2	3. d1	f1	4. e2	f2...
1. b4	d2	2. d5	c5	3. c4	e3	4. d4	e5

Puzzle 4

Red 1. c4 is the only way to prevent Blue from connecting immediately by b5, c3 or c4. Red threatens to connect c4 to the top and bottom edges with assistance from the ladder escapes at f2 and f5.

Blue 2. c5 is met by Red 3. a6, which connects to the bottom edge with assistance from the ladder escape at f5 and threatens to connect to c4 or climb to a4 and then connect to the top edge with assistance from the ladder escape at f2.

Blue may attempt a ladder escape fork by playing in the edge template between c4 and the top edge, but 2. c3 is met by 3. b4 and 2. c2 is met by 3. b3, which maintain the connection to the top edge and block Blue's ladder escape.

Other authors

Puzzle 1

There are two main solutions. In the first line, Blue at (*) on move 7 or 9 is also winning.

In the second line, Blue has several winning moves starting move 9, but we show just one example.

Surprisingly, these are the only winning first moves for Blue. For example, 1. h8 is a losing move, but the refutation is subtle and depends on play in the lower-left obtuse corner:

However, Red's refutation fails when Blue plays 1. g8 instead, because after Red 12, Blue can instead play 13. d8! at (*), which threatens to connect to g8 (which wouldn't have been possible had Blue started with 1. h8).

Puzzle 2

Note that Red is connected upwards by Template J5. The only way to prevent Red from connecting downwards is to play in the cell marked (*).

Perhaps surprisingly, (*) is a winning move, while any intrusion into the template on the first move is losing. If Red responds as expected at 2.c5, Blue's response is to intrude Red's upward connection at 3.b4. This leaves Blue in a strong position.

If Red instead plays 2.g4, Blue can respond with 3.d5, again leaving Blue in a strong position.

,

Other possible moves by Red also have a strong response, for example 2.c6 3.c4, or 2.d4 3.d5, or 2.d5 3.c4, or 2.e4 3.e5.

Puzzle 3

To simplify the analysis, it helps to note that because c9 connects to the left and a3 connects to the top, the left and top areas of the board are completely settled. Moreover, i2 captures j1 and j2 and kills h3, which in turns captures i3 and j3. Additionally, f6 and g5 are captured by Red. Therefore, the situation of the puzzle is equivalent to the following:

Therefore, whatever Red can do, Red must do in the lower right corner. Here is how Red can win:

Puzzle 4

To simplify the analysis, it helps to note that Blue's g2 group is already connected to the right edge (see below), and Red's c6 group is already connected to the bottom edge. Moreover, Blue does not have any prospects for escaping a 2nd row ladder approaching the lower left from under c6. For these reasons, the position in the puzzle is equivalent to the following:

Therefore, everything Blue can do must be done in the upper left corner. The winning move is c4:

Blue 1. c4 threatens to connect left by b5 or c3 and right by d5, d4 or e3. If Red replies in the region a2-c2-c4-a6, other than at c3, to threaten c4's connection left, then Blue 3. c3 connects to the left edge, and to the right via e2 or d5. The only reply outside the region a2-c2-c4-a6 that prevents the group of 4 blue pieces connecting immediately to the left edge by d5, d4 or e3 is Red 2. d4. However, Red 2. d4 and 2. c3 can be answered by 3. e2.

To see why Blue's g2 group is connected right, Blue's main threats are 1. g5, connecting via Edge template IV1d and by the ladder escape fork 1. i1 2. j1 3. i3 4. i2 5. g4. Red can only meet both these threats by playing in overlap of these templates, in the marked cells.

However these replies can be met by:

2. i3 3. i2 4. h2 5. g4 6. h3 7. h5

2. j3 3. i2 4. h2 5. g4 6. h3 7. h4 8. i3 9. i5

2. g4 3. i1 4. j1 5. i2 6. j2 7. i3 8. j3 9. i5 10. i4 11. h5 12. h4 13. f6

2. h4 then either 3. g4 threatening 5. i2 and 5. g6 - or simply 3. i2 4. h2 5. g4 6. h3 7. g6 - forming a crescent and connecting with the assistance of the second row ladder escape at i2

2. j2 3. h2 4. i2 (capture-dominates the alternatives i1 & j1) 5. h5 - a double ziggurat connection

[Note that although we used Edge template IV1d from g5 in the above explanation to minimise the overlap and the resulting case analysis, we don't need the whole template or the method of connection used for that. If however we use the smaller and simpler Edge template IV1a from g5, then the overlap with the ladder escape fork template is larger and we need to consider the additional replies h3, i2 and j1 (although j1 is capture-dominated by i2, so any winning variations against i2 also work against j1). Although 3. g5, connecting via Edge template IV1d, works against all these replies, there are simpler and quicker responses, e.g.: 2. h3 3. i4; 2. i2 3. h3 4. i3 5. h5 or 4.j3 5. i5; 2. j1 3. i3. So the analysis above only demonstrates that Blue is connected, rather than showing the smallest connection template or the quickest or simplest method of connection.]

Puzzle 5

Red's mustplay region consists of these 8 cells, because if Red plays anywhere else, Blue plays at d4 and wins immediately.

The unique winning move is Red c4. Red then connects for example like this:

Another move that looks plausible is Red d4, but it fails to this:

Puzzle 6

The unique winning move is c4. In the follow-up, Black must play d3 to get edge template IV2b::

Puzzle 7

The unique winning move is c6:

Indeed, now Blue is connected left by double threat at d4 or b6, and right by f6 or a ladder escape fork from f4. The uniqueness of the winning move is discussed in more detail in the article on the mustplay region.

Puzzle 8

The unique winning move is b7. In the following line, Red's moves below (1, 3, 5, and 7) are the only moves that preserve the win. Note that moves 1 and 3 are minimaxing moves, while move 7 is a foiling move that blocks the ladder escape created from Blue 6.

See Also

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