Pricing a Tree by Hand
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An exponent too large to compute directly is priced by walking it in squaring steps: multiply a few times, name the magnitude reached, and repeat.
Invent a board game — Grid — with exactly 30 legal moves in every position, and price its plans 40 moves deep. Start small: 30 squared is 900. Square again: 30 to the fourth is 900 times 900 — 810,000, call it nearly a million. So every additional 4 moves multiplies the plan count by nearly a million. Forty moves is ten such multiplications: a million, times a million, ten times over — a 1 followed by roughly fifty-nine zeros. Now weigh that against hardware: a computer checking a trillion plans per second, running since the universe began, would have examined fewer than one plan in every billion billion billion of them. The tree wins, and it is not close.
Grid is invented so the arithmetic stays exact; real games branch unevenly, and the conclusion survives every such variation.
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