# Difference between revisions of "1998 AJHSME Problems/Problem 18"

## Problem

As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at X. What does the paper look like when unfolded?

$[asy] draw((2,0)--(2,1)--(4,1)--(4,0)--cycle); draw(circle((2.25,.75),.225)); draw((2.05,.95)--(2.45,.55)); draw((2.45,.95)--(2.05,.55)); draw((0,2)--(4,2)--(4,3)--(0,3)--cycle); draw((2,2)--(2,3),dashed); draw((1.3,2.1)..(2,2.3)..(2.7,2.1),EndArrow); draw((1.3,3.1)..(2,3.3)..(2.7,3.1),EndArrow); draw((0,4)--(4,4)--(4,6)--(0,6)--cycle); draw((0,5)--(4,5),dashed); draw((-.1,4.3)..(-.3,5)..(-.1,5.7),EndArrow); draw((3.9,4.3)..(3.7,5)..(3.9,5.7),EndArrow); [/asy]$

$[asy] unitsize(5); draw((0,0)--(16,0)--(16,8)--(0,8)--cycle); draw((0,4)--(16,4),dashed); draw((8,0)--(8,8),dashed); draw(circle((1,3),.9)); draw(circle((7,7),.9)); draw(circle((15,5),.9)); draw(circle((9,1),.9)); draw((24,0)--(40,0)--(40,8)--(24,8)--cycle); draw((24,4)--(40,4),dashed); draw((32,0)--(32,8),dashed); draw(circle((31,1),.9)); draw(circle((33,1),.9)); draw(circle((31,7),.9)); draw(circle((33,7),.9)); draw((48,0)--(64,0)--(64,8)--(48,8)--cycle); draw((48,4)--(64,4),dashed); draw((56,0)--(56,8),dashed); draw(circle((49,1),.9)); draw(circle((49,7),.9)); draw(circle((63,1),.9)); draw(circle((63,7),.9)); draw((72,0)--(88,0)--(88,8)--(72,8)--cycle); draw((72,4)--(88,4),dashed); draw((80,0)--(80,8),dashed); draw(circle((79,3),.9)); draw(circle((79,5),.9)); draw(circle((81,3),.9)); draw(circle((81,5),.9)); draw((96,0)--(112,0)--(112,8)--(96,8)--cycle); draw((96,4)--(112,4),dashed); draw((104,0)--(104,8),dashed); draw(circle((97,3),.9)); draw(circle((97,5),.9)); draw(circle((111,3),.9)); draw(circle((111,5),.9)); label("(A)",(8,10),N); label("(B)",(32,10),N); label("(C)",(56,10),N); label("(D)",(80,10),N); label("(E)",(104,10),N); [/asy]$

## Solution

### Solution 1

By reversing the folds, we find that the holes will be

1) In the middle (in relation from left to right)

2) At separating poles (up and down)

This is pictured in \$\boxed{B\$ (Error compiling LaTeX. ! File ended while scanning use of \boxed.)

### Solution 2

If you unfold the paper, the hole you punched will not move. The only answer that keeps the hole in the upper-right quadrant in the same place is $\boxed{Z}$.