This is difficult to explain, but make one square flat on a table. Then place four of the remaining five matches in each corner of the square (place them vertically). Then lean the matches in adjacent corners against each other so that from one angle you've made two triangles. Then lay the fifth match with one end on the tip of each triangle. You're actually making a triangular prism with three square bases. I've never seen this riddle before, it's pretty good. _ Step 1. l_l
Step 2. /_\ on both sides
Step 3. Lay fifth match across top.
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A sort of triangle of squares. Lay out 3 squares side by side using 10 matches. Take the middle match from the bottom row and use it and the other two to make a square based on the middle match of the top row.
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A square has 4 sides therefore 3 squares from 12 toothpicks will simply be three unconnected squares
Use squares and try it out for yourself. Get a number of squares and make a rectangle 3 squares long by 4 squares wide. Count the squares. You should have 12 squares (or 3*4). That's the best way I know to prove the formula.
A cube.