A340000 Solution to stacking stones puzzle (see Comments).

1, 17, 33, 46

Offset

1,2

Comments

This is a variant of the stepping stone puzzle sequence (A337663), where you start by placing n 1's on an infinite square grid. Then place the numbers 2,3,... in order on the grid, following the rule that the sum of the 8 surrounding cells has to be equal to the number placed on a cell. a(n) is the largest number which can be achieved starting with n 1's. Additionally, there are 2 different new rules in this "stacking stones" sequence:

1. You can "stack" numbers on top of already placed numbers when the sum of the surrounding 8 cells is equal to the new number. The number inside the cell is not added to the sum and is replaced as value of the cell with the new number.

2. The starting 1's have to be at least one cell apart (to avoid a trivial infinite solution).

Links

Table of n, a(n) for n=1..4.

S. Brunner, Python program

Example

Illustration for a(3)=33:
+------+------+------+------+------+------+------+
|      |      |      |      |      |      |      |
|      |      |  18  |  17  |  26  |      |  33  |
|      |      |      |  [4] | [13] |      |      |
+------+------+------+------+------+------+------+
|      |      |      |      |      |      |      |
|      |      |      |   1  |   3  |   5  |  28  |
|      |      |      |      |      |      |      |
+------+------+------+------+------+------+------+
|      |      |      |      |      |      |      |
|      |  21  |  14  |  27  |   2  |  23  |      |
|      |      |      |  [6] |      | [11] |      |
+------+------+------+------+------+------+------+
|      |      |      |      |      |      |      |
|  22  |      |   7  |      |      |   1  |  24  |
|      |      |      |      |      |      | [12] |
+------+------+------+------+------+------+------+
|      |      |      |      |      |      |      |
|      |   1  |   8  |  31  |  32  |      |  25  |
|      |      |      | [15] | [16] |      |      |
+------+------+------+------+------+------+------+
|  30  |  29  |      |      |      |      |      |
| [20] | [19] |      |      |      |      |      |
| [10] |  [9] |      |      |      |      |      |
+------+------+------+------+------+------+------+
Illustration for a(4)=46:
+------+------+------+------+------+------+------+------+------+------+------+------+
|      |      |      |      |      |      |      |      |      |      |      |      |
|  32  |      |  37  |  44  |      |      |      |      |      |      |      |      |
|      |      | [15] | [22] |      |      |      |      |      |      |      |      |
+------+------+------+------+------+------+------+------+------+------+------+------+
|      |      |      |      |      |      |      |      |      |      |      |      |
|  24  |   8  |   7  |      |      |  39  |      |      |      |  42  |      |  29  |
|      |      |      |      |      |      |      |      |      |      |      |      |
+------+------+------+------+------+------+------+------+------+------+------+------+
|      |      |      |  31  |      |      |      |      |      |      |      |      |
|      |  16  |   1  | [13] |  18  |  21  |      |      |      |  41  |   1  |  28  |
|      |      |      |  [6] |      |      |      |      |      |      |      |      |
+------+------+------+------+------+------+------+------+------+------+------+------+
|      |      |      |      |      |      |      |      |      |      |      |      |
|      |  17  |      |   2  |   3  |      |      |  43  |      |  40  |      |  27  |
|      |      |      |      |      |      |      |      |      | [14] |      |      |
+------+------+------+------+------+------+------+------+------+------+------+------+
|      |      |      |      |      |      |      |      |      |      |      |      |
|      |  36  |  19  |      |  34  |   4  |   9  |  33  |   1  |  12  |  26  |      |
|      |      |      |      |  [1] |      |      | [10] |      |      |      |      |
+------+------+------+------+------+------+------+------+------+------+------+------+
|      |      |      |      |      |      |      |      |  46  |      |      |      |
|      |      |      |  20  |      |   5  |      |      | [23] |      |  38  |      |
|      |      |      |      |      |      |      |      | [11] |      |      |      |
+------+------+------+------+------+------+------+------+------+------+------+------+
|      |      |      |      |      |      |      |      |      |      |      |      |
|      |      |      |  45  |  25  |  30  |  35  |      |      |      |      |      |
|      |      |      |      |      |      |      |      |      |      |      |      |
+------+------+------+------+------+------+------+------+------+------+------+------+

Crossrefs

Cf. A337663.

Sequence in context: A283395 A244752 A138393 * A044062 A044443 A158057

Adjacent sequences: A339997 A339998 A339999 * A340001 A340002 A340003

Keyword

nonn,more,hard

Author

S. Brunner, Dec 26 2020

Status

approved