Problem 25 · 2011 Math Kangaroo
Stretch
Logic & Word Problems
transformationscasework
Three big boxes P, Q and R are stored in a warehouse. The upper picture on the right shows their placement seen from above. The boxes are so heavy that they can only be rotated 90° around a vertical edge, as indicated in the pictures below. Now the boxes should be rotated to stand against the wall in a certain order. Which arrangement is possible? (Choice E: all four arrangements are possible.)
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Answer: B
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Hint 1 of 3
Each allowed move is a 90° turn about a vertical edge, which both slides a box and rotates its top label a quarter turn.
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Hint 2 of 3
So a box's final orientation is tied to how far it travelled: track position and label-rotation together.
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Hint 3 of 3
Test each pictured line-up and reject any whose letters point the wrong way for the moves needed to reach it.
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Approach: couple each box's final position to its forced label orientation
- Tipping a box 90° about a vertical edge moves it one step and turns its label a quarter turn, so position and orientation change together — they cannot be set independently.
- Walking the boxes to the wall one move at a time, each box's letter must end pointing the way that number of quarter-turns dictates.
- Comparing the four pictured line-ups, only one has every letter oriented consistently with the moves that put the boxes there.
- That reachable arrangement is choice (B).
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