A340000 Solution to stacking stones puzzle (see Comments).

1, 17, 33, 46

Offset: 1

Author

S. Brunner, Dec 26 2020

Keywords

• nonn
• more
• hard

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).

Examples

			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  |      |      |      |      |      |
|      |      |      |      |      |      |      |      |      |      |      |      |
+------+------+------+------+------+------+------+------+------+------+------+------+
		

Links

• S. Brunner, Python program

Crossrefs

Cf. A337663.