Cube with One Edge Sliced Off
AKA Recycle Box
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Task Design and construct the following solid:  Here is the top view:  Solution (A TI-34 calculator is used below.) Look at the small right triangle in the corner of the square (top view):  2a2 = 4 a2 = 2 a = √2 The area of the small triangle is 1/2*base*height = 1/2*√2*√2 = 1 square inch. The volume of the slice that is cut off of the cube (i.e., the area of the triangular prism) is (area of the triangle)*height = 1*x = x cubic inches. The volume of the whole (intact) cube is x3. We need to find x. We know the volume V of our solid is x3 - x = 30 cubic inches. So we need to solve the equation x3 - x - 30 = 0 for x. We use the fixed point method. Rewrite x3 - x -30 = 0 as x3 = x + 30 Then 3√ x3 = 3√(x + 30), or x = 3√(x + 30) The algorithm is: repeat: 3√(x + 30) → x In the program, x is shown on the display and stored in Ans. Program OP1 = 3√(Ans + 30) Now enter a first guess for x, which becomes Ans, for example, 4. 4 Enter Now continually press OP1. You will see:
Check that x = 3.21446795 satisfies the equation x3 - x - 30 = 0 as follows: STO → A A3 - A - 30 = Enter You will see 0. So x = 3.21446795 ≈ 3 3.5/16 inches, and x - √2 ≈ 1.800 ≈ 1 13/16 inches. We are now ready to design our plan for the box. It has five lateral sides which are rectangles, and a top and bottom which are congruent pentagons. Here is one possible plan, using 3 pieces.   Here are some pictures of a recycle box: 
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