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A352621 a(n) is the minimum possible value of the largest number placed in a solution to the Forest of Numbers (Bosque de Números) puzzle if we start with the numbers 1 and 2 in an n X n grid (see Comments).

%I A352621 #36 Jan 09 2023 02:40:51
%S A352621 6,12,36,68,140
%N A352621 a(n) is the minimum possible value of the largest number placed in a solution to the Forest of Numbers (Bosque de Números) puzzle if we start with the numbers 1 and 2 in an n X n grid (see Comments).
%C A352621 Start with an n X n square grid. Each cell has up to eight neighbors. Place the numbers 1 and 2 in any two cells of the grid. Then place a number in each remaining cell, in increasing order (not necessarily using consecutive integers), and with the rule that when you place the number k in a cell, the sum of the numbers in its neighboring cells must equal k. The goal is to fill the grid in such a way as to minimize the largest number placed.
%C A352621 This is similar to the Stepping Stones problem discussed in A337663, but predates it by more than 20 years.
%C A352621 Computer solutions by _Dmitry Kamenetsky_.
%C A352621 a(7) = 292, a(8) = 502, a(9) = 787 and a(10) 1391 not yet confirmed to be optimal.
%H A352621 Rodolfo Kurchan, <a href="https://www.puzzlefun.online/problems">Puzzle Fun Forest of Numbers</a>
%e A352621 4 X 4 = 36 solution by _Rodolfo Kurchan_:
%e A352621   +---+---+---+---+
%e A352621   | 33| 21|  5| 28|
%e A352621   +---+---+---+---+
%e A352621   | 11|  1|  4| 19|
%e A352621   +---+---+---+---+
%e A352621   | 24| 10|  3|  7|
%e A352621   +---+---+---+---+
%e A352621   | 36|  2| 22| 32|
%e A352621   +---+---+---+---+
%e A352621 3 is sum of 1+2, 4=1+3, 5=1+4, 7=3+4, 10=1+2+3+4, 11=1+10, 19=3+4+5+7, 21=1+4+5+11, 22=2+3+7+10.
%e A352621 .
%e A352621 5 X 5 = 68 solution by _Dmitry Kamenetsky_:
%e A352621   +---+---+---+---+---+
%e A352621   | 32| 18| 63| 30| 56|
%e A352621   +---+---+---+---+---+
%e A352621   | 13|  1|  4| 10| 16|
%e A352621   +---+---+---+---+---+
%e A352621   |  9|  3| 39|  6| 54|
%e A352621   +---+---+---+---+---+
%e A352621   | 36|  5|  2|  8| 14|
%e A352621   +---+---+---+---+---+
%e A352621   | 12|  7| 22| 46| 68|
%e A352621   +---+---+---+---+---+
%Y A352621 Cf. A337663, A350627.
%K A352621 nonn,more
%O A352621 2,1
%A A352621 _Rodolfo Kurchan_, Mar 24 2022