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 Bouncing Bishops (Posted on 2021-04-15)
Below is a representation of a 7x12 chessboard with two bishops present, one at the location marked Z and one at the location marked 0. Each makes successive simultaneous single-unit moves, with 0 starting down and to the right and Z starting down and to the left. Their first six moves are marked 1, 2, ... and A, B, ... in the diagram.

As you can see, each continues in its same direction until a barrier is hit--either the edge of the board or one of the X's, as shown in the diagram. Imagine each X is a block of rubber that occupies its full square, and each bishop's diameter matches the whole width of its square, so that it bumps into the rubber block on the next square over.

If going toward a corner, either of the board or where the edge of the board meets an X square, the bishop starts to retrace its steps. The same would happen if the bishop were to be aiming directly for a square occupied by a block--retrace its steps backward.

```
· · · · · · · · · · · ·
· · · · · · · · · · · ·
· · · F · · X · · · · ·
· · E · · · 5 · · · · X
· D · Z · 6 · 4 · 0 · ·
C · A · · · · · 3 · 1 X
· B · · X · · · · 2 · ·

```

On what numbered move would the two bishops share the same square? ... and at what row and column would this meeting take place?

Bonus 1: If the two bishops survive meeting on one square and continue on their respective ways, where else would they meet?

Bonus 2: On what move would both be back in their starting position going in the same direction as at the beginning?

Bonus 3 (lower difficulty): On some other puzzle of this type, what would be the possible numbers of places on the bishop's full route where the bishop does reverse his path?

 See The Solution Submitted by Charlie Rating: 3.0000 (1 votes)

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 Solution | Comment 1 of 4
The two bishops would meet on move number 17.  Numbering the rows 0-6 from top to bottom, and the columns 0-11 from left to right, they would meet in square [r,c] = [5,6], moving in opposite directions.

Bonus 1: If allowed to continue, they would meet again on moves 54, 60, 67, 74, and 80 in squares [4,3], [2,3], [3,10], [2,3] and [4,3] respectively.

Bonus 2: They will both return to their original orientations on the 100th move.

Bonus 3: The bishops could reverse their path in a corner square, or on the edge of the board next to an X, or in a square diagonally adjacent to an X. There are 17 such squares. However any particular bishop's path would only include two of them, as their path would simply bounce back and forth between them.

 Posted by tomarken on 2021-04-15 08:24:08

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