Problem 2 · AMC 8 Stretch
Core
Counting & Probability
Geometry & Measurement
account-for-all-possibilitiesorganizing-datasymmetry
Look at a six-pointed star (a Star of David) built from a triangular grid. Hidden inside are triangles of three different sizes — some point up and some point down. How many triangles are there in all? (This is a classic 'don't miss any!' counting puzzle.)
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Answer: 20 triangles
Show hints
Hint 1 of 4
First decide how many different SIZES of triangle you can find. There are three.
Still stuck? Show hint 2 →
Hint 2 of 4
For each size, count the up-pointing ones and the down-pointing ones separately. A great trick: cut a cardboard triangle of each size and slide it around so you don't miss any.
Still stuck? Show hint 3 →
Hint 3 of 4
The star looks the same flipped top-to-bottom, so for each size the number pointing up equals the number pointing down.
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Approach: Sort by size and direction, then add
- Two skills are needed: seeing that there are three sizes, and counting carefully so none get missed.
- Because the star looks the same flipped upside down, for each size the 'up' count equals the 'down' count.
- Tally the three sizes:
Size Up Down Total Small 6 6 12 Medium 3 3 6 Large 1 1 2 - Add the totals: 12 + 6 + 2 = 20.
- So there are 20 triangles in all (12 small, 6 medium, 2 large). The big idea: when a puzzle says 'count them all,' get organized instead of randomly pointing and hoping.
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