# Create a 15-X puzzle with a unique solution

## Background

Last week, Brandan came up with a genre about filling a 6x6 grid with polystrips of lengths 1 through 8, using each length exactly once. A polystrip cannot bend twice in a row to cover a 2x2 region. The clues are given as endpoints of the strips, 15 in total. In the original version, the endpoints were marked with symbols so that each pair of identical symbols indicates a strip (one symbol given only once indicates a monomino):

. . . . A B
A C . . . .
D E F . . .
. . . F . .
. D . E C .
G H H B . .


In an answer, I proposed a variation where two kinds of symbols are given so that each strip connects from A to B (there are one more As than Bs; one of the As indicates a monomino):

A A A B . A
A B . . B A
. . B . . .
. B . . . .
. B . A . .
. . . A B .


What happens when you replace A and B with just a single character X? So you need to join X to X. Does that still have a unique solution? An interesting puzzle in itself is to find one arrangement of X that DOES have a unique solution.

And I did find one. Now it's your turn.

## Challenge

Construct a "Fifteen Crosses" puzzle that has a unique solution. Rules of the puzzle:

• A 6x6 grid is given. Fifteen cells are marked with an X.
• Use polystrips of lengths 1, 2, ..., 8 exactly once each to cover the 6x6 grid. All Xs must be at the endpoint of one of the strips. Polystrips cannot bend twice in a row to cover a 2x2 region.
• Love it! Thanks for making this puzzle for me :) Commented Mar 14 at 13:46
• Great post! I've been offline lately doodling with the A/B version of this puzzle and have found it quite addicting. I want to try to turn it into a little app. This was my original intent: find something simple but interesting enough to motivate me to learn the tools I need to publish a game. Commented Mar 18 at 18:47
• As a mathematical aside, I came to this shape (1 thru 8 in a 6x6) while studying triangular numbers and their intersection with near-squares. (Obviously you could do this with other rectangles that aren't near-squares.) This is really the only non-trivial human-puzzlable size except perhaps the next biggest. The list is 1:3 (2x3), 1:8 (6x6), 1:20 (14x15), 1:49 (35x35) and continues to explode from there. Commented Mar 18 at 19:00

The following puzzle has a unique solution

X X X X X X
. . . . . X
. . . . . X
. . . . . X
. . . . X X
X X . . X X


Explanation of solution path

Notice that in the sixth column, the first, fifth and sixth Xs must belong to either the monomino or the domino so we can place these right away. This forces the remaining Xs in the first row and the sixth column to expand downwards or to the left, respectively. The rest of the polyominoes quickly fall into place proceeding from the top right diagonally downwards.

• Nice! The idea of constraining the smaller ones first like that never occurred to me. Commented Mar 12 at 22:52
• Great solution using matching Commented Mar 14 at 13:49

This puzzle has unique solution:

• Pretty close, but I see 2 solutions. The 2 and 3 paths can drawn in 2 ways. Commented Mar 13 at 20:19
• Florian is right. I think this one can be made unique by moving the X at the top left corner to the 2nd row, 2nd column. Commented Mar 13 at 23:32
• Yeah you are right! Commented Mar 14 at 6:45