Can you cover the surface of a cube with 16 identical colourful rectangles? No overlappings are allowed and the cube has to be fully covered.
The cube from Example 3 is a present and one layer of a gift-wrap is totally not enough. Can you cover it with another 15 identical rectangles? You can assume the covering from Example 3 was thin and it did not affect the shape of a cube. As before no overlappings are allowed and the surface has to be fully covered by rectangles.
Can a \(5\times5\) square checkerboard be covered by \(1\times2\) dominoes?
Cut an equilateral triangle into 4 smaller equilateral triangles. Then can another equilateral triangle be cut into 7 smaller equilateral triangles (triangles do not necessarily have to be identical)?
Consider another equilateral triangle. Is it possible to cut it into (a) 9; (b) 16; (c) 28; (d) 2; (e) 42 smaller equilateral triangles (which are not necessarily identical)?
(f) Kyle claims he can cut an equilateral triangle into any number of smaller (not necessarily identical) equilateral triangles if this number is either greater than 8 and divisible by 3, or greater than 3 and has remainder 1 when divided by 3. Prove or disprove Kyle’s statement.
(g)* Let \(n\) be a natural number greater than 5. Is it true one can cut an equilateral triangle into \(n\) smaller equilateral triangles?
a) What is the answer in case we are asked to split the figure below into \(1\times4\) rectangles instead of \(1\times5\) rectangles?
(b) In the context of Example 1 what is the answer in case we are asked to split the figure into \(1\times7\) rectangles instead of \(1\times5\) rectangles?
Can you cover a \(10 \times 10\) board using only \(T\)-shaped tetraminoes?
Can you cover a \(10 \times 10\) square with \(1 \times 4\) rectangles?
Two opposite corners were removed from an \(8 \times 8\) chessboard. Can you cover this chessboard with \(1 \times 2\) rectangular blocks?
One small square of a \(10 \times 10\) square was removed. Can you cover the rest of it with 3-square \(L\)-shaped blocks?