# Difference between revisions of "2003 AMC 12A Problems/Problem 13"

m (2003 AMC 12A/Problem 13 moved to 2003 AMC 12A Problems/Problem 13) |
m |
||

Line 1: | Line 1: | ||

== Problem == | == Problem == | ||

− | The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attatched to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? | + | The [[polygon]] enclosed by the solid lines in the figure consists of 4 [[congruent]] [[square (geometry) | squares]] joined [[edge]]-to-edge. One more congruent square is attatched to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a [[cube (geometry) | cube]] with one face missing? |

[[Image:2003amc10a10.gif]] | [[Image:2003amc10a10.gif]] | ||

Line 7: | Line 7: | ||

== Solution == | == Solution == | ||

− | Let the squares be labeled <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>. | + | Let the squares be labeled <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math>. {{image}} |

When the polygon is folded, the "right" edge of square <math>A</math> becomes adjacent to the "bottom edge" of square <math>C</math>, and the "bottom" edge of square <math>A</math> becomes adjacent to the "bottom" edge of square <math>D</math>. | When the polygon is folded, the "right" edge of square <math>A</math> becomes adjacent to the "bottom edge" of square <math>C</math>, and the "bottom" edge of square <math>A</math> becomes adjacent to the "bottom" edge of square <math>D</math>. |

## Revision as of 10:24, 11 November 2006

## Problem

The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attatched to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?

## Solution

Let the squares be labeled , , , and . ----
*An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.*

When the polygon is folded, the "right" edge of square becomes adjacent to the "bottom edge" of square , and the "bottom" edge of square becomes adjacent to the "bottom" edge of square .

So, any "new" square that is attatched to those edges will prevent the polygon from becoming a cube with one face missing.

Therefore, squares , , and will prevent the polygon from becoming a cube with one face missing.

Squares , , , , , and will allow the polygon to become a cube with one face missing when folded.

Thus the answer is .