Problems & Puzzles: Puzzles Puzzle 1.- The Gordon Lee puzzle You are asked to fill a matrix nxn with numbers (0 --> 9) and to count the distinct primes embedded in the matrix, regarding that you can read the lines or part of them, in form vertical, horizontal or diagonal orientation, in both directions. This puzzle was first proposed (1989) by Gordon Lee for a 6x6 matrix, and answered by Stephen C. Root, of Westboro, Mass. 317333 ( Recently Marcus Oswald produced 5 more solutions, all of them with 187 primes ) http://www.geocities.com/MotorCity/7983/primesearch.html Jaime Ayala and Carlos Rivera extended the Gordon Lee puzzle to other size of matrixes and obtained the current records : 113 1139 11933 http://www.astro.virginia.edu/~eww6n/math/PrimeArray.html Would you like to try to beat those records ? Solution This 13/04/99 - almost a year after we post the Gordon Lee puzzle - James Bonfield from the Medical Research Council - Laboratory of Molecular Biology, Cambridge, England, came up with other solutions to the 4x4 matrixes that make even with the current record for this size of matrixes: 63 primes 1139 7739 3973 *** Well, this is the month of the Gordon Lee Puzzle. At 17/04/99, Wilfred Whiteside, from Houston Tx., wrote: "I found a second prime (5x5) array with 116
primes" For the 7x7 case he has found the following matrix with 276 primes: 1 9 3 1 3 3 3 Regarding the prediction of the number of primes in each kind of matrix, he comments: "It is interesting that a simple power law fudge formula 1.626n^2.6475 comes within plus minus 1 of the number of primes for n=1 to n=6 and predicts that for n=7 you would have around 280 primes and around 400 primes for the n=8 case". And adds "My gut feeling is that 276 wouldn't hold up long as a record if it were posted, not with guys like Marcus Oswald out there. My 276 result is probably analagous to finding a 6x6 with 184 primes. So I'm guessing that 280 primes is probably about where the upper limit is". If you think that this is too much work... wait to see the next Whiteside results: "For fun [Bolds are mine, CR] I've started playing around with 3-D arrays. So far I have found 5 3x3x3 arrays that have 82 primes in them (if my program is working correctly - it is hard to have full confidence in it without anyone else's results to compare). The 5 3-D arrays are: 7 3 3 7 8 4 9 6 7 9 7 1 8 1 1 3 3 7 9 7 1 3 8 6 7 4 3 9 7 1 3 8 6 7 4 3 9 7 1 3 8 6 7 4 3 It is intersting that all the record 2-D matrixes have no zeros in them (which I guess isn't all that suprising), but so far my best 3-D 3x3x3's all have a central zero which produces enough 3 digit primes that have their second digit equal to zero to outweigh the loss of two digit primes that its presence causes. If my program is testing correctly, there are [n(3n-2)]^2 numbers that have to be pulled out of a nxnxn array and tested for primality and uniqueness. For the 3x3x3 case this means 441 numbers have to be pulled out and tested as opposed to the 65 tests for a 3x3 array. Thats a lot of numbers to test for such a small array and of course it gets far worse very quickly as n is increased. I doubt I will ever explore beyond the 4x4x4 case. Any doubth that just "for fun" is a powerful string Wilfred Whiteside has finished his study of the 3x3x3 matrix and at 24/04/99 wrote his results: "The 8 unique arrays found so far (all containing 82 primes) are listed below. 1 1 3 7 6 3 4 6 7 1 1 3 5 3 8 7 9 3 1 1 3 4 5 7 9 7 1 1 1 3 4 5 7 9 7 1 1 1 3 4 5 7 9 7 1 1 1 3 4 5 7 9 7 1
1 1 3 4 5 7 9 7 1 1 1 7 7 1 3 9 8 3 Wilfred Whiteside, at 29/04/99 improved his own record for the 7x7 matrix jumping from 276 to 281 primes. "Here is the 7x7 array with 281 primes (closely matches that fudged polynomial I made up): 9 3 3 7 3 1 3 At (8/5/99) Eric W. Weisstein wrote: "I have proved by exhaustive enumeration that the 30-prime solution to the 3x3 prime array is unique (modulo rotation and reflection) and that no 3x3 solution with more primes exists". Alberto Hernández Narváez, from Monterrey, México, sent (22/07/99) the following 8x8 record matrix: .1 3 1 6 3 3
9 3 Embedded primes= 373 Alberto got this matrix developing a generalized code based in another made by me time ago for the 6x6 original problem. He ran this new code over an initial matrix that contains the 281 primes 7x7 solution of Whiteside in certain position of the matrix (shown in coloured numbers) and zeros at the first row and column. 373 primes are a kind of close from the 400 predicted by Whiteside for this level of matrixes. BTW, Whiteside himeself has confirmed on request the primality of the 373 solution sent by Alberto. Wilfred Whiteside, at 29/04/99 improved the Hernández solution for the 8x8 matrix. This is the new record: 3 9 9 7 3 1 9 2 The last October 31, Wilfred Whiteside improved his better solution for the 8x8 matrix. Now he got a solution with 382 primes inside. 3 3 9 3 1 3 1 9 Please notice that "There were 10 even digits in this
Gordon matrix". 3911 3391 1477
9941 *** His last solution (13/03/2000) contains 283 primes:
1 9 3 3 7 1 7 7 1 0 3 1 1 1 3 3 *** Now, Marcus Oswald has beatten the Whiteside's record for the 8x8 (382 primes embedded), producing 3 solutions having 385 primes embedded. This is his email: "...during the weekend my computer found three 8x8 matrices including 385 primes. If there is no error in my testing routine this would be a new record for a 8x8 matrix. The three matrices are: 1 3 5 1 1 9 3 3 3 3 1 9 9 7 3 9 3 3 1 9 9 7 3 9 The second matrix has even a 0-digit in it! And the second and the third matrix only differ in this single digit. *** Much later (May 2003) he (Marcus) wrote again: I also found new best solutions for the 8x8 problem, 3 grids with 386 primes and 6 grids with 387 primes: 3 3 3 7 1 9 9 3 386 3 3 3 7 1 9 9 3 386 3 3 3 2 9 3 3 9 386 3 3 3 9 1 7 3 9 387 1 3 9 9 9 1 3 3 387 1 3 3 2 7 1 3 3 387 1 3 9 9 9 1 3 3 387 3 3 3 7 1 9 9 3 387 3 3 3 7 1 9 9 3 387 The next problem I want to attack (if I have a little bit more time) is to prove the optimum value of 63 for the 4x4 problem. *** Wilfred Whiteside got a better solution (390 primes versus 387 primes, the better record by M. Oswald) sent on August 18, 2003:
So, the competition is still alive... *** On March 2005, this first puzzle of my pages is still alive and well alive, indeed!. Mike Oakes wrote:
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So, the new 8x8 record in from Whiteside: 394 primes. Congratulations Wilfred! *** |
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