OFFSET
1,2
COMMENTS
This is a slightly simplified version of the problem as originally stated in the Le Monde video. For the original version, see A221866.
The problem is as follows (we state it for the 3 X 3 case).
Draw a 3 X 3 square and write "1" in the upper left corner-square. Fill in the other squares one by one according to these rules:
- select an empty square
- write in it the sum of its neighboring squares (a "corner-square" has 3 neighbors, an "edge-square" 5 and the "center-square" 8)
- when all squares are marked, record the highest written value.
Example:
+-----+-----+-----+
| 1 | | |
| | | |
+-----+-----+-----+
| | | |
| | | |
+-----+-----+-----+
| | | |
| | | |
+-----+-----+-----+
+-----+-----+-----+
| 1 | | |
| | | |
+-----+-----+-----+
| | | |
| | | |
+-----+-----+-----+
| | | 0 |
| | | |
+-----+-----+-----+
+-----+-----+-----+
| 1 | 1 | |
| | | |
+-----+-----+-----+
| | | |
| | | |
+-----+-----+-----+
| | | 0 |
| | | |
+-----+-----+-----+
+-----+-----+-----+
| 1 | 1 | |
| | | |
+-----+-----+-----+
| | 2 | |
| | | |
+-----+-----+-----+
| | | 0 |
| | | |
+-----+-----+-----+
+-----+-----+-----+
| 1 | 1 | 3 |
| | | |
+-----+-----+-----+
| | 2 | |
| | | |
+-----+-----+-----+
| | | 0 |
| | | |
+-----+-----+-----+
+-----+-----+-----+
| 1 | 1 | 3 |
| | | |
+-----+-----+-----+
| | 2 | 6 |
| | | |
+-----+-----+-----+
| | | 0 |
| | | |
+-----+-----+-----+
+-----+-----+-----+
| 1 | 1 | 3 |
| | | |
+-----+-----+-----+
| | 2 | 6 |
| | | |
+-----+-----+-----+
| | 8 | 0 |
| | | |
+-----+-----+-----+
+-----+-----+-----+
| 1 | 1 | 3 |
| | | |
+-----+-----+-----+
| | 2 | 6 |
| | | |
+-----+-----+-----+
| 10 | 8 | 0 |
| | | |
+-----+-----+-----+
+-----+-----+-----+
| 1 | 1 | 3 |
| | | |
+-----+-----+-----+
| 22 | 2 | 6 | <--- MAX value = 22 (but 44 is possible)
| | | |
+-----+-----+-----+
| 10 | 8 | 0 |
| | | |
+-----+-----+-----+
REFERENCES
Eric Angelini and others, A Grid With MAX, Postings to the Math Fun Mailing List, April 14 2013 onwards.
LINKS
Phil Carmody, Gridsplode [A sandbox for experimenting]
EXAMPLE
From Hans Havermann, Apr 13 2013: (Start)
a(3) is 44.
If we label the 9 squares 1-9:
+-----+-----+-----+
| 1 | 2 | 3 |
| | | |
+-----+-----+-----+
| 4 | 5 | 6 |
| | | |
+-----+-----+-----+
| 7 | 8 | 9 |
| | | |
+-----+-----+-----+
then one can obtain 44 by filling in the squares in the order:
{2, 4, 7, 5, 3, 6, 8, 9} or
{2, 4, 7, 5, 3, 6, 9, 8}
or their symmetrical counterparts
{4, 2, 3, 5, 7, 8, 6, 9} or
{4, 2, 3, 5, 7, 8, 9, 6}
(End)
***********************************************
From Fred W. Helenius, Apr 14 2013: (Start)
a(4) = 2473:
The filled-in grid looks like this:
1 1 1239 2473
2 4 419 815
6 18 100 296
6 30 48 148
It is easy to figure out the order in which the numbers are entered.
(End)
************************************************
From Fred W. Helenius, Apr 14 2013 and Christian Boyer, Apr 15 2013: (Start)
a(5) = 297136:
There are four solutions, one of which is
1 1 40 73 235
2 6 33 162 470
2 10 16 1313 632
148574 49530 16947 3906 1945
297136 99032 32555 11702 5851
(End)
************************************************
The current record for a(6) is 128493518 from Christian Boyer, Apr 16 2013.
************************************************
Further lower bounds from Christian Boyer, Apr 18 2013: (Start)
Here is a better 7 X 7 grid, with 139015458134, using a similar strategy to my 6 X 6:
1 2 6 18 54 162 162
1 4 12 36 108 486 810
213342341 1066711707 1706738772 72104324924 139015458134 1404 2700
213342336 640027013 3486463028 17151921630 49759201338 8208 4104
140356800 72985536 4232225497 7726494333 24880751139 24624 12312
42107040 25264224 5614272 1871424 320112 110808 36936
8421408 8421408 2807136 935712 615600 184680 36936
************************************************
If I continue my "snail" strategy on 8 X 8, I obtain 541048181546137:
1 2 6 18 54
162 486 486
1 4 12 36 108
324 1458 2430
17280729221 86403646107 288012153716 437778473776 1313335421640
1313335452366 4212 8100
17280729216 51842187653 149766319904 875556947396 3940006295178
6566677271892 24624 12312
5760243072 11520486144 271062231376625 91645561810642 21888924401648
10506683887182 73872 36936
3789633600 1970609472 541048181546137 178326013015414 64791307846516
32395629305414 221616 110808
1136890080 682134048 151585344 50528448 16842816
2881008 997272 332424
227378016 227378016 75792672 25264224 8421408
5540400 1662120 332424
It is easy to continue this on larger grids, but is this "snail" strategy the best one?
If so the game is less interesting.
(End)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane, Apr 17 2013
STATUS
approved