Problem 4 · 2023 Math Kangaroo
Easy
Spatial & Visual Reasoning
cube-views
Nine steps of a staircase winding around a cylinder can be seen, starting at the bottom and leading all the way to the top. All the steps are equally high. How many steps cannot be seen?
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Answer: D — 12
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Hint 1 of 3
From the front you only see the steps facing you; the rest of the staircase keeps winding around the hidden back of the cylinder.
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Hint 2 of 3
The picture shows the front-facing steps spiralling up; figure out how high the whole tower climbs, then take away the nine you can already see.
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Hint 3 of 3
Each level of the spiral has steps on both the front and the back, so the hidden back steps roughly mirror the visible front ones, plus a few more for the extra height.
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Approach: see that the spiral has front and back steps at each level, then count the hidden ones
- The nine steps you can see are the ones facing you as the staircase spirals up the front of the cylinder.
- As the spiral turns, the same number of steps run around the hidden back at each level, and the tower keeps climbing past where the front steps stop.
- Counting the back steps level by level all the way to the top gives twelve steps that are turned away and cannot be seen.
- So the number that cannot be seen is D, 12.
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