# Reverse puzzling - What is my brother solving?

This puzzle is inspired by the previous reverse puzzles

My brother and I love puzzles. But I know he is solving some puzzles without me. So yesterday I sneaked into his bedroom to find out what he is working on. The only thing I found is this piece of paper with numbers on it :

• (2,8)
• (2,1) (2,3) (7,8) (9,8)
• (2,4) (2,6) (4,8) (6,8) (7,1) (7,3) (9,1) (9,3)
• (1,8) (2,7) (2,9) (3,8) (4,1) (4,3) (6,1) (6,3) (7,4) (7,6) (9,4) (9,6)
• (1,3) (3,1) (4,6) (6,4) (7,9) (9,7)
• (1,4) (3,6) (4,7) (6,9) (7,2) (8,1) (8,3) (9,2)
• (1,9) (3,7) (4,2) (6,2) (8,4) (8,6)
• (1,2) (1,6) (3,2) (3,4) (4,9) (6,7) (8,7) (8,9)
• (1,7) (3,9) (8,2)

Can you help me identify what puzzle it is ?

Hint 1:

When I was in my brother's room I have seen a kind of grid. I don't remember it exactly but it was way smaller than a 9x9 grid.

• what's your brother's name? or do we have to guess? – JMP Jun 2 '16 at 20:57
• @JonMarkPerry his name doesn't matter – Fabich Jun 2 '16 at 20:58
• you've created a new tag? why not provide description to it? – klm123 Jun 6 '16 at 11:28
• Perhaps because you need 20k reputation to do that. I would guess that LoD has in fact provided a description and it's waiting in a queue for someone with more administrative privileges to put it in place. – Gareth McCaughan Jun 6 '16 at 11:38
• PS. I added the tag to the other 4 reverse-puzzling puzzles. – Gareth McCaughan Jun 6 '16 at 11:38

## 7 Answers

The puzzle he's solving is

Given a white knight and a black knight in the middle of opposite sides of a 3x3 chessboard, swap them in the least amount of moves. The pairs of the numbers mean:

Number the squares as follows: The first number in each pair is the location of the black knight, the second is the location of the white knight. The nth line lists the positions reachable in n-1 moves, so the first line is the initial position (2,8), the second line lists the positions reachable in one move: (2,1), (2,3), (7,8), (8,9), and so on.

The last pair written is (8,2), which is the position in which both knights have swapped places.

When drawn on a hexagonal grid, it looks symmetric (2,8) red

(2,1) (2,3) (7,8) (9,8) green

(2,4) (2,6) (4,8) (6,8) (7,1) (7,3) (9,1) (9,3) blue

(1,8) (2,7) (2,9) (3,8) (4,1) (4,3) (6,1) (6,3) (7,4) (7,6) (9,4) (9,6) yellow

(1,3) (3,1) (4,6) (6,4) (7,9) (9,7) magenta

(1,4) (3,6) (4,7) (6,9) (7,2) (8,1) (8,3) (9,2) cyan

(1,9) (3,7) (4,2) (6,2) (8,4) (8,6) black

(1,2) (1,6) (3,2) (3,4) (4,9) (6,7) (8,7) (8,9) dark red

(1,7) (3,9) (8,2) dark green

• Hint1 added !!! – Fabich Jun 6 '16 at 8:48

Colored grid: red,orange,light green,dark green,light blue,dark blue,grey,pink,purple

• Bah, you just beat me to it – Tony Ruth Jun 2 '16 at 20:53
• Same. Mine was numbers and with row-to-column instead of column-to-row – Bulldogg6404 Jun 2 '16 at 20:54
• Hint 1 added !! – Fabich Jun 6 '16 at 8:49

OK, let me state the (now) obvious:

The puzzle is somehow based on a 3x3 grid. Each list of pairs indicates a bunch of somehow-related squares on that grid, the squares being numbered 1 to 9. (In what follows I'll assume the numbering is in "raster order": 123 on top row, 456 on middle row, 789 on bottom row, reading left to right.)

But

the central square (5) is not used.

The pairs

are ordered, perhaps indicating that what we have is lists of possible moves from square to square. Every pair of (different) squares appears once each way around.

Each group of pairs

has a lot of symmetry to it. Specifically, let A be what you get by reflecting about a vertical axis, and let B be what you get by reflecting about a horizontal axis and then reversing the order of all the pairs; then applying either of these leaves all the groups unchanged (though of course reordered).

My current best guess is that this describes

some kind of game or puzzle in which you move something around a 3x3 grid

but I don't like this guess and suspect something a little more interesting is going on. Perhaps

there are pieces on many of the squares and you're supposed to make them swap places or something like that.

Note that

neither of those is possible if the stages are meant to describe successive moves in the puzzle, since the first three things listed all have the form (2,x). (Unless we start with at least three things on square 2!)

I also remark that

if you have knights on a 3x3 chessboard, all the 8 squares around the edge are mutually accessible and the middle square isn't -- but I don't see anything very knight's-move-y about this puzzle so I suspect there's some other explanation for the fact that the central square appears to be going unused.

There's another obvious way (or family of obvious ways) to do the numbering:

draw a 3x3 magic square with numbers 1..9, and use those numbers.

If you do this,

the moves in each step become more consistent in direction. Specifically, if your magic square looks like 816/357/492 then the first four groups have all the "broadly SE-to-NW" moves, the fifth has the three "narrowly SW/NE" ones in both directions, and the last four have the "broadly NW-to-SE" moves. That's cute but I don't currently see its significance (if any).

Further,

suppose that in that magic square you number squares by "northwestness": 432/321/210. Then the first group contains the unique move that changes this by +4; the next contains the four that change it by +3; the next contains the eight that change it by +2; the next contains the twelve that change it by +1; the next contains the three that don't change it at all. This pattern fails for the remaining groups, though.

This puzzle continues to make me feel stupid.

• "I don't see anything very knight's-move-y about this puzzle" - are you sure about that? A knight in square 8 can move to 1 or 3. – ffao Jun 7 '16 at 16:32
• Some of the pairs correspond to knight's moves, for sure. But that's just because every pair that doesn't involve the number 5 is represented. – Gareth McCaughan Jun 7 '16 at 17:27
• To be more explicit, suppose (2,8) corresponds to a pair of knights at 2 and 8. The positions of these two knights after one move are exactly the pairs on the second line. – ffao Jun 7 '16 at 17:29
• That's an interesting observation. – Gareth McCaughan Jun 7 '16 at 18:58
• OK, so suppose (2,8) means you have a white knight on 2 and a black knight on 8. Then it looks (I haven't checked every detail) as if each successive line shows what positions can be attained after 1,2,... single knight-moves with, I assume, the restriction that we never list a position we've already listed (i.e., if we number from 0, line n shows precisely the positions that are attainable after n knight-moves starting with (W2,B8)). So then the table shows that it takes 8 moves to swap the knights, and that nothing takes longer. – Gareth McCaughan Jun 7 '16 at 19:02

Same post, different format...

I find it easier to compare similarities with the spaces marked in number form instead of colored. Note the board I was using has the vertical axis labeled going downward, so the pattern may be flipped from existing answers (in case it makes a difference). Maybe he is preparing

some kind of next pattern question. If you line up two sets of numbers from 1 to 9 and connect the sets to each other with lines based on the ordered pairs (creating graphs that look a bit like lacing up a shoe) the graphs for each line in the problem is symmetrical. In relation to the first hint two sets lined up are't a specifically a grid but if they were they could be 2x9. That's what I have so far, but it probably doesn't contribute much.