# Summing numbers and then increasing used numbers

I was making a mathematical game (viewable here: www.michaelprimo.it/beta) where you have a 3x3 grid, one central yellow button with a generated number and the other eight blue buttons with clickable and summable numbers. I thought: "what if everything is predictable?" So I made this:

• There are eight numbers: 1, 1, 1, 1, 1, 1, 1, 1, and a main number of value 2
• You must sum the numbers you have to reach exactly that number
• After that, all the numbers used for the sum AND the main number will be increased by 1
• The next move will have 2, 2, 1, 1, 1, 1, 1, 1, and 3 as a main number. Now we must select numbers to make a sum of 3.
• This is repeated until you finish the moves

What is the biggest main number you can reach? More importantly, is the game doable (as in no particular tricks and it's not broken in some way)

I want to know if this puzzle I accidentally created during my game is good or not. I am not good with game theory, but I am interested in the argument.

• Welcome to Puzzling! What exactly do you mean by "doable"? What kinds of tricks are you worried about? How would it be "broken"? Jan 3 '21 at 0:23
• Thank you! Well, maybe the game can be infinite by repeating a pattern and this shouldn't be happened. I tried and I am trying myself to solve it too and I don't think this will happen, but I decided to ask help to people more smarter than me for this task because I don't think to can do it now. Jan 3 '21 at 0:28
• So you're asking what the highest possible main number is, or if a strategy/proof exists that would guarantee no highest possible number exists? Jan 3 '21 at 0:29
• Actually both, but I am more interested of the first question. If the second question is true then the main number can be infinite, but if there is a proof the puzzle is not broken by some pattern then the main number can reach a finite limit and we need to know what is that limit. Jan 3 '21 at 0:31
• If the starting (playable) squares contained (2,1,1,1) instead of (1,1,1,...) then it is easy to reach any number. Jan 3 '21 at 17:18

The biggest number you can reach is

38

This is because

I ran an exhaustive brute force search to play all possible games. (Disclaimer: this python code is just a quick hack, there are better ways to implement this.)

from sympy.utilities.iterables import multiset_partitions

for K in range(1,9):
G = [[2]+[1 for i in range(K)]]
m = 2
game_states = set()
while True:
game = G.pop(-1)
if tuple(game) not in game_states:
if sum(game[1:])==game[0]:
G.append([i+1 for i in game])
if G[-1][0]>m:
m=G[-1][0]
else:
for msp in multiset_partitions(game[1:],2):
if sum(msp[0])==game[0]:
G.append(sorted([game[0]+1]+[i+1 for i in msp[0]]+msp[1],reverse=True))
elif sum(msp[1])==game[0]:
G.append(sorted([game[0]+1]+[i+1 for i in msp[1]]+msp[0],reverse=True))
if len(G)>0 and G[-1][0]>m:
m=G[-1][0]
if len(G)==0:
break
print(K,m)


As a bonus, highest reachable numbers a(k) on k=1,2,...,8 tiles are:

k a(k)
1 2
2 3
3 5
4 5
5 8
6 11
7 17
8 38


• Thank you so much for your help! I done 38 before, so I guess I reached without knowing the limit of the puzzle without knowing it. At least we know now it's not flawed and have a solution. Thank you again! Jan 3 '21 at 17:24
• @MichaelPrimo The general problem could be interesting. On 9 tiles, the biggest reachable number is 468 (or more). It appears this sequence grows very quickly. Jan 3 '21 at 17:28
• That is interesting, I don't know why since I can't describe what kind of puzzle/problem is this. Maybe I can think about it when I will put it to the game I am making (in fact the puzzle is an alternative to the game I made here: www.michaelprimo.it , without the beta. ) The yellow number is generated random this time and you have a time limit in fact the puzzle was born as a "what if" and then I liked it, I was curious about finding more and here I am. Jan 3 '21 at 17:37