IBM Ponder This #271 · November 2020

The following game is played with a list of natural numbers (greater than 1). At each turn, one of the following two operations is performed:

1. The numbers a,b are removed from the list and the number a^b (a to the power of b) is added to the list.
2. A number of the form a^b (where a,b>1) is removed from the list, and the numbers a,b are added to the list.

Note that the list can contain multiple instances of the same number.

A game can be described by a sequence of lists, each obtained from the previous one by one of the operations above. As an example consider the following list:

64
8,2
2,2,3
8,2
256

The above list demonstrates how, starting with a list containing 64, a list containing 256 can be obtained via four steps (although two steps suffice).

To avoid writing large numbers explicitly, we can also use the shorthand notation for powers:

8^2
8,2
2,2,3
2^3,2
2^2^3

Your goal: Demonstrate a game that within 20 steps reaches a list containing the number 2147483647 (the Mersenne prime discovered by Euler). The initial list is constrained as follows:1. It cannot contain more than five numbers. 2. All the numbers except one should be at most 50. 3. The remaining number can be at most 1,000,000,000 Provide your game solution in the exact format presented above.

A bonus '*' for explaining (without giving the full game) how 2147483647 can be reached from a list containing the number 64 alone. More than 20 steps can be used.

Solution

Best opened after a real attempt

To be added.