Latest Update of this Site: August 30, 2023

Collatz Prizes offered by Ingo Althofer

The following problem, first stated by Lothar Collatz back in 1937, is wellknown:

Start with an odd number n. Build 3n+1, and then half on this until finally an odd number is reached again. Repeat this loop until either 1 is reached or until you lose interest. The Collatz conjecture says that for each starting value n after finitely many steps 1 is reached. A formal proof is still missing. We want to motivate people to think more deeply on this problem and some variants. Therefore, we offer a few money prizes.

* The 5n+1 problem is almost identical to the 3n+1 problem. The only difference is that instead of 3n+1 the expression 5n+1 is built. Analogously, in the X*n+1 problem the expression 3*n+1 is substituted by X*n+1.

Prize 1

Prove or disprove that there exists some odd number X larger or equal to 5 and an odd number n(0), such that starting value n(0) leads to infinity under the rule n(t+1) = X*n(t) + 1 and subsequent halfing. (25 Euro)

Prize 2

Prove or disprove that n(0) = 1 leads to infinity under the rule n(t+1)= 9*n(t) + 1 and subsequent halfing. (50 Euro)

Prize 3

This prize concerns a 2-player game with alternating move order.

The current state in the game is an odd number n. The player to move builds either 3n+1 or 3n-1. Then iterative halfing is done, until an odd number is reached again. If this new number is 1, the player has won the game. If not, the other player is to move. Observe: the player to move has only the choice between two options.

Computer analysis by Michael Hartisch has shown that in case of optimal play by both players, the game will end in 1, when started in a position with a number n < 1 million.

A prize of 500 Euro is offered for a proof that all odd starting numbers n will lead to 1, if both players act optimally.

For short, we call the game "the 3n+-1 game". The game was created in July 2023 by Ingo Althofer.

Main Prize

Prove or disprove the original Collatz conjecture:
All n(0) lead to 1 after finitely many steps in the 3n+1 problem. (1,000 Euro)


Prizes only for solutions submitted until December 31, 2037.
Prizes only for first solutions for that question.
Legal actions are excluded.

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