Math Puzzle: Mysterious pyramid
Can you ascend this unusual pyramid?
The puzzle involves nine cells arranged in a triangle, each containing a zero. The objective is to change the numbers to nine consecutive positive integers starting with a, where a is the smallest integer. A move consists of increasing or decreasing the values of two adjacent cells by 1. The puzzle begins with the sum of the numbers in the orange cells equal to the sum in the black cells, initially both being zero. Throughout the puzzle, this equilibrium remains unchanged.
At the end, the numbers should be a, a+1, ..., a+8. Since the sum of the six smallest numbers (a, a+1, a+2, a+3, a+4, a+5) must be less than or equal to the sum of the three largest numbers (a+6, a+7, a+8), it follows that 6a+15 ≤ 3a+21. Solving this inequality yields a ≤ 2.
However, the total sum of the numbers 1 through 9 is 45, which is odd. For the sums of the orange and black cells to remain equal (r=b), this sum must be even. Therefore, if a solution exists, it can only occur for a = 2. The puzzle can be solved through a series of moves, as demonstrated in the accompanying image.
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