A003213 Number of ways to quarter a 2n X 2n chessboard.

1, 1, 5, 37, 782, 44240

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• more

Comments

Warning: it now seems very likely that this is an incorrect version of A257952. - N. J. A. Sloane, Apr 17 2016

Number of ways to dissect a 2n X 2n chessboard into 4 congruent pieces.

One can ask the same question for a 2n+1 X 2n+1 board if one omits the center square: this gives A006067.

a(0)=1, since there is one way to do nothing.

Comment from Andrew Howroyd, Apr 18 2016: (Start)

This sequence is wrong because of a bug in Mr. Parkin's code, and amazingly I can pinpoint exactly what the bug is! (I can reproduce his results.)

Firstly the description of the problem and its solution in Mr. Parkin's letter is very clear -- he doesn't leave a lot of room for misinterpretation (this is hugely to his credit). He also includes a very clear description of his algorithm, so I decided I would just code it up. I obtained Giovanni Resta's results as given in A257952 -- there is nothing wrong with Mr Parkin's algorithm.

A detailed breakdown of Parkin's results is also provided in the letter. All the results match with the exception of the final line. (This would be highly improbable if there was a completely different interpretation.) In any case, one sentence stood out as a possible red flag: "Further, there are potential mirror image paths in both cases when starting on the centre lines and these are prevented by requiring a turn in one direction on the path prior to allowing a turn in the other direction" (bottom of page 6). The discrepancy in results does indeed relate to the center line and if I modify my code to lose the flag on recursion, then I get Mr. Parkin's results (so turn in one direction is only prohibited for one step). (End)

References

• M. Gardner, The Unexpected Hanging and Other Mathematical Diversions. Simon and Schuster, NY, 1969, p. 189.
• Popular Computing (Calabasas, CA), Vol. 1 (No. 7, 1973), Problem 15, front cover and page 2.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. R. Parkin, Letter to N. J. A. Sloane, Feb 01, 1974. This letter contained as an attachment the following 11-page letter to Fred Gruenberger.
• T. R. Parkin, Letter to Fred Gruenberger, Jan 29, 1974, Page 1.
• T. R. Parkin, Letter to Fred Gruenberger, Jan 29, 1974, Page 2.
• T. R. Parkin, Letter to Fred Gruenberger, Jan 29, 1974, Page 3.
• T. R. Parkin, Letter to Fred Gruenberger, Jan 29, 1974, Page 4.
• T. R. Parkin, Letter to Fred Gruenberger, Jan 29, 1974, Page 5.
• T. R. Parkin, Letter to Fred Gruenberger, Jan 29, 1974, Page 6.
• T. R. Parkin, Letter to Fred Gruenberger, Jan 29, 1974, Page 7.
• T. R. Parkin, Letter to Fred Gruenberger, Jan 29, 1974, Page 8.
• T. R. Parkin, Letter to Fred Gruenberger, Jan 29, 1974, Page 9.
• T. R. Parkin, Letter to Fred Gruenberger, Jan 29, 1974, Page 10.
• T. R. Parkin, Letter to Fred Gruenberger, Jan 29, 1974, Page 11.
• T. R. Parkin, Discussion of Problem 15, Popular Computing (Calabasas, CA), Vol. 2, Number 15 (June 1974), page PC15-4.
• T. R. Parkin, Discussion of Problem 15, Popular Computing (Calabasas, CA), Vol. 2, Number 15 (June 1974), page PC15-5.
• T. R. Parkin, Discussion of Problem 15, Popular Computing (Calabasas, CA), Vol. 2, Number 15 (June 1974), page PC15-6.
• T. R. Parkin, Discussion of Problem 15, Popular Computing (Calabasas, CA), Vol. 2, Number 15 (June 1974), page PC15-7.
• T. R. Parkin, Discussion of Problem 15, Popular Computing (Calabasas, CA), Vol. 2, Number 15 (June 1974), page PC15-8.
• Popular Computing (Calabasas, CA), Illustration showing that a(3) = 37, Vol. 1 (No. 7, 1973), front cover. (One of the 37 is simply the square divided into four quadrants.)

Crossrefs

Bisection of A006067. Cf. A064941.

See A257952 for another version.