The peg problem asks whether a round peg in a square hole fits better than a square peg in a round hole. This compares the ratio of the area
of a circle to its circumscribed square with the ratio of the area
of a 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).
This result holds in dimensions 
. For 
, the 
-hypercube fits more closely into the 
-ball than vice versa, while the two
ratios are equal for 
(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)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
is the gamma
function. The ratios in question are then
 |  |  |
(4)
|
 |  |  |
(5)
|
(Singmaster 1964). The ratio of these ratios is
![(R_(round peg))/(R_(square peg))=(pi^nn^(n/2))/(2^(2n)[Gamma(1+1/2n)]^2),](/images/equations/Peg/NumberedEquation1.svg) |
(6)
|
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 
.
