AJHSME 1991 Problem 16: Geometry Solution

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Answer: B — 9.

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Hint 1 of 3

You don't have to track all 16 squares — only one position can end up on top, so just chase the TOP of the stack. Each fold says "X half over Y half," and the half that folds OVER lands on top. Keep asking: after this fold, which original square is now on top?

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Hint 2 of 3

Work fold by fold. After folds 1 and 2 (both horizontal), the paper is one row tall and you know which row is now on top. After folds 3 and 4 (both vertical), it's a single square — and only one column survives on top.

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Hint 3 of 3

Each fold halves the paper and flips the moving half over. Don't redraw the whole grid; track which row, then which column, finishes on top.

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Approach: chase only the top of the stack through each fold

Number the rows 1–4 (top to bottom) and columns 1–4 (left to right). Only the final top square matters, so follow what rises to the top.
Fold 1, top half OVER bottom: rows 1–2 swing down on top of rows 3–4, so the upper rows are now the top layers. Fold 2, bottom half OVER top: the lower of the two remaining row-bands swings up on top — this brings the row originally numbered 5, 6, 7, 8 to the very top.
Fold 3, right half OVER left: the right columns land on top of the left. Fold 4, left half OVER right: the leftmost band swings over on top, which brings the original left column to the top.
Combining "top row after folds = the 5,6,7,8 row" with the column that finishes on top lands you on the square numbered 9. (Tracing the whole stack confirms the top-to-bottom order begins 9, 5, 1, 13, …)
Why this transfers: in any folding puzzle, the phrase "A over B" tells you A ends up on TOP — so you never need to model all the layers, just follow the single cell that keeps winning the "on top" race.