Difference between revisions of "Solutions to puzzles"

Latest revision as of 14:12, 25 August 2024

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.

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.

Eric Demer

Puzzle 1

The unique winning move is red a3. a2 is captured to the top edge, and Red threatens b4. e4 is a ladder escape, so b3 loses to a4 and b5 loses to b4. b4 capture-dominates a4 and a5. Thus Blue plays b4.

Red responds with e2. This connects to the bottom edge by edge template IV2d, and connects to the top edge as shown.

Uniqueness:

The shaded area is a virtual connection from c3, so Red must either play in it or play e2.

The rest of this paragraph uses edge template III2a extensively, and so we do not mention it each time it's used. If Red plays a5 or b4, then Blue plays b3, winning via d3 or e1. If Red plays a4, then Blue plays c4, connecting left via a3 or a5. a3 is Red's winning move. If Red plays b3, then Blue plays c2, connecting left via a3 or b4. If Red plays c2, then Blue plays c4, winning.

Red's only remaining try is e2. Blue responds with c4. After that, consider edge template III2a to the left edge.

If Red intrudes there other than at *, then Blue responding at * captures those 6 cells. If Red intrudes at * before Red plays c2, then Blue responding at c2 still captures the template. Thus, until Red has played c2, Red has no useful way of even threatening to disconnect c3 from the left edge, so one gets

At this point c3+c4 escapes the ladder from b2.

Puzzle 2

The unique winning move is blue d2.

This connects to the left edge via edge template IV2g, so Red must play either in there or e2. Against most intrusions in that template, Blue just responds with c3. Specifically, c3 almost makes this a second order template, since even if Red starts with two moves in it, the only way Red can stop Blue from reconnecting is if those moves are in the two cells marked "*":

Specifically, Red must have defended against b4. If Red did that with b4 or a5, then Blue reconnects with b3 or c1. Thus, Red needs a4. Red must have also defended against b3. If Red did that with b3 or a3, then Blue reconnects with c1. Thus, Red also needs b2.

Since Blue c3 connects to the right edge via edge template III2b, Red must play one of the shaded cells in the following diagram.

Blue responds to e2 with d4. Whether before or after that exchange, Blue responds to each of b2,a4 with b3. At least two of b2,a3,a4 would still be empty, so that allows Blue to climb, for example like this:

Before the e2,d4 exchange, Blue responds to c3 with d4. After the e2,d4 exchange, Blue can respond to c3 with d3. In each case, Blue will have d2 and d4 captured to the right edge, so Red must defend against c1. If Red does so with b2, then b3 is an escape fork for the ladder from b5. If Red does so anywhere else, then a2 still escapes that ladder.

Thus, both before and immediately after the e2,d4 exchange, none of Red's intrusions into the edge template IV2g work, so Red must try e2 followed by d3. However, against that, Blue just plays b5, and ladders to a2.

Uniqueness:

Red's main threat is d3, bridging to edge template III2d. d2 is Blue's winning move. e2 and d3 lose to c3, since d5 escapes the ladder. d4 and c5 lose to b4, forming edge template IV2g.

If Blue plays c4, then Red plays b4. b4 will be captured to the bottom edge, so Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3.

This leaves just b5. Red reponds with b4.

That has a connection to the bottom edge, via a5 or c4, and threatens c2. If Blue plays c4, then Red plays a5, killing b5 and thereby reverting to the "Blue plays c4, then Red plays b4" case. If Blue intrudes in the other three cells of the connection from b4 to the bottom edge, then Red plays c4, re-establishing that connection and connecting to the top edge via edge template IV2a. Thus, Blue must defend against c2. c3 loses to b3, since e1 escapes the ladder. The other four all lose to d3, since that connects via d4 or a5.

Puzzle 3

The unique winning move is blue b2.

That captures a2 and a3, and threatens e1, so Red must play in c1,c2,e1. The b2-d1 bridge is bolstered at c1 (by Red's edge), so c1 is now worse than c2.

If Red plays c2, then Blue plays d2. That captures e1 and e2, so Red must respond with c1. Blue then plays b4, which connects left via e5 or b3, and connects right via edge template IV2e.

Thus Red plays e1. Blue responds with e2.

Red must play in c1,c2,e1. As before, we discard c1. If Red plays c2, then Blue plays d2, killing e1 and thereby reverting to the "Red plays c2, then Blue plays d2" case. Thus Red plays d2. Blue responds with c4, connecting left via a5 or c3 and connecting right via e3 or d5.

Uniqueness:

I use it twice here, in already-dense prose, so I start with what I will call "the c4 ziggurat-defense":

If Blue plays in the e3 ziggurat other than at c4, then Red can respond at c4, after which Red can keep both c4 and e3 connected down inside the ziggurat: Either c4 is captured down, in which case e3 connects via one of {d3,d4,e4}, or Red can defend the bridge, in which case c4+e3 connects down via one of {b5,c5,e4}.

With that out of the way, suppose Blue plays anywhere other than the five cells marked "*" in the following diagram.

Red responds with b2, capturing b1 and c1. Blue gets one more move before it becomes Red's turn again, and Blue's two moves can't be Blue occupying both c4 and a5.

One of Blue's two moves must be in {a5,a3,b3}, and the other of Blue's two moves must defend against d2, since red d2 would connect down via edge template IV2i.

If that other move from Blue is not in the e3 ziggurat, then Red wins by laddering to e3: If Blue played a5 then Red plays b4, else Red defends the a4-b2 bridge.

If Blue played in the e3 ziggurat other than at c4, then Red uses the c4 ziggurat-defense, after which Red wins with d2 or the ladder along the bottom. Thus, one of Blue's two moves must be c4. This means Blue's other move can't be a5, so it must have been a3 or b3, to defend against red a5. In either case, Red plays b4, winning via either d2 or the other of a3,b3.

Thus, Blue must have played one of the five cells marked "*" in the following diagram.

b2 is Blue's winning move, and against the other four, Red plays d2.

If Blue has played b1 or c1, then we claim that the entire red-shaded region below is captured by Red:

To prove this, it is sufficient to show that Red has a strategy in the shaded region that keeps both a4 and d2 connected to the bottom edge. Because by the time the board is completely filled, if a4 and d2 are connected to the bottom edge, no minimal blue winning path can pass through the shaded area. Therefore all blue stones in that area will be dead.

To show that a4 and d2 are simultaneously connected down, consider any blue move in the region. If Blue plays b4, then Red responds with a5. In that case, d2 still connects down via edge template IV2i. Thus, Blue tries somewhere else in this region. Red responds at b4, and Blue gets one more move before it becomes Red's turn again.

If Blue's two moves were a5 and b5, then Red plays d4, connecting both d2 and b4 to the bottom with a trapezoid. Otherwise, Blue played at most one of a5,b5. If Blue played exactly one of those two, then Red plays the other, killing whichever of those two was Blue. In that case, Blue again gets one more move before it goes back to being Red's turn, so for showing this capture, we are left with the case where neither of Blue's was was in {a5,b5}. Since b4 is red, a5 and b5 are directly red-captured, so one of Blue's two moves must be c3, since otherwise Red plays there. The order in which Blue played Blue's two moves doesn't matter, so one can pretend Blue played c3 first. In that case, before Blue played Blue's other move, Red had a bridge to e3 and a connection from e3 due to the ladder escape at b4. Thus, Red can respond to restore the connection down from d2.

Due to this capture, the positions simplify to the following,

where the shading indicates that one of b1,c1 is blue and the other is empty.

Blue must play e1. After that, if c1 is empty, Red plays at c1 and is connected by a crescent, and if b1 is empty, Red plays at a2 and is connected by a different crescent and edge template II. This shows that both 1. b1 and 1. c1 are losing for Blue.

The remaining moves for Blue are a5 and c4. If Blue plays a5, then we claim that within the red-shaded region in the following diagram, Red can keep d2 connected down while ensuring that red b4 would connect down.

The proof is similar to our proof of capture above. Specifically, before Blue plays there, d2 has a template down and red b4 would connect down since e3 escapes the ladder. If Blue plays b4, then red b4 can't happen and d2 still connects down, since b4 isn't in the template. If Blue plays c3, then Red plays d4, capturing d2 down and guaranteeing a ladder escape for red b4. The d2-e3 bridge is bolstered at e2, so e2 is worse than d3. If Red plays in the e3 ziggurat other than at c4, then Red uses the c4 ziggurat-defense, after which d2 connects down via c3 or e2, and red b4 would also connect down. If Blue plays c4, then Red plays d4. That captures d2 down, so red b4 would connect down via b5 or c3.

Given that, Red wins as follows:

Thus​ 1. a5​ loses, so Blue's only remaining try is ​1. c4.

Red threatens

and

so Blue must play one of the cells shaded below.

If Blue plays in {d5,d3,e2}, then Red plays b4, which captures a5 and b5, so Blue must defend against b2. Blue playing in {a3,b2,b1} loses to c2, Blue playing c1 loses to b3 - threatening e1 and a2 - and Blue playing b3 loses to b2, since that connects d2 to the top via e1 or c2.

If Blue plays e1 or c2, then Red plays

and then wins with whichever of e1,c2 Blue did not play.

If Blue plays c1, then Red plays b3 and defends that bridge. That connects to the top with a2 or e1 and to the bottom with a5 or d4.

If Blue plays b3, then Red plays b2. That captures b2 and d2 to the top, so Red wins with b4 or d4.

Lastly, if Blue plays a5, then Red wins as follows.

Puzzle 4

Uniqueness:

Blue's main threat is c2, bridging to edge template III2a.

If Red plays in {b1,b2,a2,a3}, then Blue plays b4, and wins with edge template IV2g. If Red plays a4, then Blue plays b2, and wins with d1 or c3.

If Red plays b3 or c2, then Blue plays b2, forcing d1.

Blue continues with c3. This connects right via edge template III2d, and connects left with c2 or b3 or b4.

If Red plays d1, then Blue plays c3. That connects right with edge template III2d and connects left with a ziggurat.

There is only 1 remaining try for Red, and that is Red's winning move: d2

Showing that d2 wins for Red:

This part is quite similar to showing that d2 wins for Blue in puzzle 2.

The following region is a virtual connection:

d3 fails to c3, since e5 escapes the ladder, c3 and c4 fail to e3, forming edge template III2d, and everything else fails to c4, connecting down via one of b5,c5,d4.

Against most intrusions in that virtual connection, Red just responds with c3. Specifically, c3 almost makes this a second order connection, since even if Blue starts with two moves in it, the only way Blue can stop Red from reconnecting is if those moves are in the two cells marked "*":

Specifically, Blue must have defended against b4. Blue's other move must have been in {c4,d4,d3}, since otherwise Red plays d4 and connects with one of c5,d5,e4. d3 fails to c4, since e5 escapes the ladder, and c4 fails to e3, forming edge template III2d, so Blue's other move must be d4. This means a5 and b4 fail to c4, so Blue also needs b5.

Since red c3 connects to the top via d1 or b2, Blue must play one of the shaded cells in the following diagram.

Red responds to d1 with b2. Whether before or after that exchange, Red responds to each of b5,d4 with c4. c4 would connect down - via b5 or c5 or laddering to e5 - so that allows Red to climb, for example like this:

Before the d1,b2 exchange, Red responds to c3 with b2. After the d1,b2 exchange, Red can respond to c3 with c2. In each case, Red will have b2 and d2 captured to the top edge, so Blue must defend against both e3 - forming edge template III2d - and the ladder escape fork at c4.

The overlap between those threats is {d3,c5}. Against d3, Red plays a4 and ladders to e5. Against c5, Red plays b5, followed by a4 or d3. (d3 connects with c4 or either of d4,e4).

Thus, both before and immediately after the d1,b2 exchange, none of Blue's intrusions into

work. Since b2 will be captured to a1, this means Blue must try d1 followed by c2. However, against that, Red just plays a4, and ladders to e5.

Puzzle 5

To find Red's winning move, let's use mustplay analysis to narrow down the possibilities. Blue has several threats, which are shown with their respective carriers:

Bridge to ziggurat:

b4 and double threat:

b4 and a different double threat:

Ladder and climbing from b2:

c4 with edge template III2b and ladder:

Red must play in the intersection of these carriers, i.e., at a4, a5, or b4.

Red a4 loses to Blue c3, with double threats towards both edges:

Red a5 loses to Blue c2 as follows. Apart from attacking Blue's bridge, which Blue can simply defend, Red's moves 3 and 5 are forced, after which Blue wins by double threat:

This leaves b4 as the only possible move for Red.

Showing that b4 wins for Red:

If Blue plays to the upper-left of the short diagonal other than in {c2,b3}, then Red responds with c2 and defends the bridge. c2 connects to the top via d1 or c1 or b2, and connects to the bottom with a very short switchback.

If Blue plays to the lower-right of the short diagonal other than at d4, then Red plays a5. This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with Edge template III2d. Thus Blue plays b3 instead. Red responds with d2, and a1 escapes that ladder.

Blue must continue with c3, and what's left of this case is checking that Red can connect e2 down no matter which of the above shaded cells was Blue's first move. If that move was neither d3 nor e3, then Red bridges to d4, and connects down with c4 or c5 or d5. Otherwise, Red plays whichever of d3,e3 Blue didn't play. Since Red has a5 and b4, b5 is dead, so Red ladders to b4.

If Blue plays d4, then Red plays c4, capturing a5 and c5. This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with Edge template III2d. If Blue plays b3, then Red plays d2, bridging from c4 and laddering to a1.

This leaves the following tries for Blue.

c2 loses to a3, using Edge template III2d

and a switchback, as shown above.

If Blue plays b3, then Red plays d2, and a1 escapes that ladder. Towards the bottom, if Blue plays outside of {c4,c3,d3}, then Red can play c4 and defend the bridge from d2, and if Blue plays outside of {d4,d5,c5}, then Red can play d4 and defend the c5+d5 capture.

In the former case, Red connects down with a5 or c5 or e4. In the latter case, Red connects down with e3 or d3 or c3, since c3 threatens c4 and a5. The sets {c4,c3,d3} and {d4,d5,c5} are disjoint, so Red gets at least one of those, and thereby connects down for a win.

If Blue plays c3, then Red plays d4, capturing e2 and b4 to the bottom. Blue must respond in {d2,c2,b3}, since otherwise Red plays c2, connecting down with d2 or b3 and connecting to the top with d1 or c1 or b2. If Blue plays d2 or c2, then Red plays a3, winning with Edge template III2d. If Blue plays b3, then Red plays d2, and wins by laddering to a1.

If Blue plays d2, then Red plays c3, connecting down with a5 or d4. Towards the top, blue c2 loses to a3, forming Edge template III2d, and Blue's other moves all lose to red c2, connecting with d1 or c1 or b2.

Blue's only remaining try is a5. Red responds to that with d4, capturing c5 and d5.

Red's main threat is d4, laddering to a1.

If Blue plays in {a2,b1,b2,c1,d1}, then Red plays c2. This connects to the top with d1 or c1 or b2, and connects down with a box.

If Blue plays c2, then Red plays d2. This connects down with a trapezoid, and connects to the top with d1 or a3, since a3 forms Edge template III2d.

If Blue plays d2, then Red plays c3. This has a bridge down, and connects to the top for the same reason as without the a5-b4 exchange.

The e2-d4 bridge is bolstered at e3 (by Blue's edge), so e3 is no better for Blue than d3.

Thus, the only remaining case is ​ ​ ​ 4. d3 ​ .

Against that, Red plays ​ ​ ​ 5. c4 ​ .

This threatens c2, which would connect to the top via d1 or c1 or b2, so Blue must respond in {c2,c3,b3}. If Blue plays c2 or c3, then Red plays a3, winning with Edge template III2d. Thus Blue plays b3 instead. Red responds with d2, connecting down via c3 or e3, and connecting to the top because a1 escapes the ladder.

Therefore ​ 1. b4 ​ does win for Red.

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.

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: