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Each volume the contains problems at the beginning and (brief) solutions at the end. As two authors have put it, "there is a general consensus among mathematicians that the two-volume Pólya-Szegő is the best written and most useful problem book in the history of mathematics." [1]: 59
A cushion filled with stuffing. In geometry, the paper bag problem or teabag problem is to calculate the maximum possible inflated volume of an initially flat sealed rectangular bag which has the same shape as a cushion or pillow, made out of two pieces of material which can bend but not stretch.
List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface;
Reprint of 1935 edition. A problem on page 101 describes the shape formed by a sphere with a cylinder removed as a "napkin ring" and asks for a proof that the volume is the same as that of a sphere with diameter equal to the length of the hole. Pólya, George (1990), Mathematics and Plausible Reasoning, Vol.
Other examples are doubling the cube and trisecting the angle. Two polyhedra are called scissors-congruent if the first can be cut into finitely many polyhedral pieces that can be reassembled to yield the second. Any two scissors-congruent polyhedra have the same volume. Hilbert asks about the converse.
If there is more than one constraint (for example, both a volume limit and a weight limit, where the volume and weight of each item are not related), we get the multiple-constrained knapsack problem, multidimensional knapsack problem, or m-dimensional knapsack problem. (Note, "dimension" here does not refer to the shape of any items.)