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1998 AJHSME Problems/Problem 18

Revision as of 21:37, 19 May 2017 by Greenivy (talk | contribs) (→‎Solution 1)

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 $\boxed{B}$

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 $\boxed{Z}$.

See also

1998 AJHSME (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

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