THE CHINESE CHESSBOARD.
(
Chessboard Problems)
Into how large a number of different pieces may the chessboard be cut
(by cuts along the lines only), no two pieces being exactly alike?
Remember that the arrangement of black and white constitutes a
difference. Thus, a single black square will be different from a single
white square, a row of three containing two white squares will differ
from a row of three containing two black, and so on. If two pieces
cannot be placed on the table so as to be exactly alike, they count as
different. And as the back of the board is plain, the pieces cannot be
turned over.
Answer:
+===I===+===+===+===I===+===+===+
| |:::: 2 ::::| 3 |:::| 5 |:6:|
I...+===+...+===+...I...I...+===I
|:::: 1 |:::| ::::| 4 |:::| 7 |
I...+===+===+...I===I...I===+===I
| |:::: |:::| ::::| 9 |:::|
I===I...I===============I...I...I
|:::: 11|:::: ::::: 10|:::| 8 |
I=======I===I===========I...I...I
| ::::: 12|:::: 13::::| |:::|
I=======+...I...+===+===|===+===I
|:::: 14|:::| |:::| 16::::| 17|
I...+...I===I===+...+...+===+...I
| ::::| ::::: 15|:::| ::::|
I=======+===========+===+=======I
|:::: ::::: 18::::: ::::: |
+===+===+===+===+===+===+===+===+
+===+===I===I===+===I===+===+===+
| ::::| |:::: |:::| ::::|
I...+===I...I=======I...I===+...I
|:::| |:::: |:::: |:::| |
I...I===I===============I===I...I
| |:::: ::::| ::::: |:::|
I===I=======I=======I=======I===I
|:::| ::::| ::::| ::::| |
I...I===+...I...+...I...+===+...I
| ::::| |:::: |:::| ::::|
I...+===I...+===I===+...I===+...I
|:::| |:::: |:::: |:::| |
I===I...+=======I=======+...I===I
| |:::: ::::| ::::: |:::|
I...+=======+...I...+=======+...I
|:::: ::::| |:::| ::::: |
+===+===+===+===+===+===+===+===+
Eighteen is the maximum number of pieces. I give two solutions. The
numbered diagram is so cut that the eighteenth piece has the largest
area--eight squares--that is possible under the conditions. The second
diagram was prepared under the added condition that no piece should
contain more than five squares.
No. 74 in _The Canterbury Puzzles_ shows how to cut the board into
twelve pieces, all different, each containing five squares, with one
square piece of four squares.
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