# Introducing: Sudoku-Janpu

This is an entry into the 29th fortnightly topic challenge - Retrograde Analysis.

## Scenario:

Alice and Bob were playing a game of Sudoku-Janpu.

Alice: I win.
Bob: (looking at the board) Oh man, not again!
Alice: You could have won before; I made a mistake.
Bob: Really?
Alice: (pointing at the board several times) See, if you went there instead of there I'd have had no legal move ... I should have gone there instead of there.
Bob: Aw... that makes it even worse! Want another game? This time I'm going to win.
Alice: (chuckles) We'll see about that!

This was the final position:

For you to work out:

How did Alice win? How could Bob have won?

Both answers require proof.

## Rules:

Sudoku-Janpu is a 2-player game played on a blank sudoku grid. Janpu means jump in Japanese.

• The first player can play anywhere
• After that, players have to play exactly a knight's jump away from the last play
• A play consists of writing a number in a blank square, as long as the number is not in the same row, column or 3x3 box
• The first person who cannot make a valid play loses
• @boboquack thanks for clarifying; I'll delete my comment that provoked the clarification lest it now confuse readers :-). Commented Mar 24, 2017 at 23:34
• I reread and think I understand it now, nevermind! Commented Mar 24, 2017 at 23:42
• And if anyone wants to turn this game into a KotH on PPCG, go ahead! Someone's probably come up with this game before me. Commented Mar 25, 2017 at 3:50

Alice's last move must have been

the 9 in the top left corner

because

every other occupied square has a knight's-move neighbour empty square that can be filled legally. (Almost all have one that can be filled with a 1, in fact.)

Clearly Bob's previous move was

the 4 lying southeast of that 9.

Now consider

the square at the top left of the centre-left 3x3 box. Call it S. This has four neighbours, all of them squares Alice moves to at some point. After she moves to the last of these, Bob can move to S, at which point Alice has no vacant square to move to and Bob wins; and this is what Alice is referring to when she says Bob could have won.

We can't

infer very much about which of those she actually moves to last, or when -- quite a number of different sequences of moves are consistent with the diagram in the question -- but after some discussion with @boboquack I have confirmed that the above is what he was wanting to see :-).

• Perhaps I am. I haven't found a better way that actually proves anything yet, though :-). Commented Mar 25, 2017 at 3:17
• And the fact that there are viable game-histories for each of Alice's next-to-last moves seems to me to show that there can't be a really simple proof. But, again, maybe I'm missing something obvious. Commented Mar 25, 2017 at 3:18
• (note to readers: earlier versions of my answer were much more complicated because I misunderstood how specific an answer OP was looking for.) Commented Mar 25, 2017 at 3:55