The answer to the question "which fits better, a round peg in a square hole, or a square peg in a round hole?" can be interpreted as asking which is larger, the ratio of the area of a circle to its circumscribed square, or the area of the square to its circumscribed circle? In two dimensions, the ratios are and , respectively. Therefore, a round peg fits better into a square hole than a square peg fits into a round hole (Wells 1986, p. 74).
However, this result is true only in dimensions , and for , the unit -hypercube fits more closely into the -hypersphere than vice versa (Singmaster 1964; Wells 1986, p. 74). This can be demonstrated by noting that the formulas for the content of the unit -ball, the content of its circumscribed hypercube, and the content of its inscribed hypercube are given by
(1)
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(2)
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(3)
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The ratios in question are then
(4)
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(5)
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(Singmaster 1964). The ratio of these ratios is the transcendental equation
(6)
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illustrated above, where the dimension has been treated as a continuous quantity. This ratio crosses 1 at the value (OEIS A127454), which must be determined numerically. As a result, a round peg fits better into a square hole than a square peg fits into a round hole only for integer dimensions .