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During the next 5 + 1 ⁄ 2 years, until June 1986, Gardner wrote 9 more columns, bringing his total to 297. During this period other authors wrote most of the columns. In 1981, Gardner's column alternated with a new column by Douglas Hofstadter called "Metamagical Themas" (an anagram of "Mathematical Games"). [1] The table below lists Gardner ...
Because the square of a standard normal distribution is the chi-squared distribution with one degree of freedom, the probability of a result such as 1 heads in 10 trials can be approximated either by using the normal distribution directly, or the chi-squared distribution for the normalised, squared difference between observed and expected value.
Where and are the cdf and pdf of the corresponding random variables. Then Y = X 2 ∼ χ 1 2 . {\displaystyle Y=X^{2}\sim \chi _{1}^{2}.} Alternative proof directly using the change of variable formula
Download as PDF; Printable version ... In the middle square, rows 1 and 2 and rows 3 and 4 have been swapped. ... 3+23, 6+20, etc. In the finished square, 1 is placed ...
When expressed as exponents, the geometric series is: 2 0 + 2 1 + 2 2 + 2 3 + ... and so forth, up to 2 63. The base of each exponentiation, "2", expresses the doubling at each square, while the exponents represent the position of each square (0 for the first square, 1 for the second, and so on.). The number of grains is the 64th Mersenne number.
Consequently, a square number is also triangular if and only if + is square, that is, there are numbers and such that =. This is an instance of the Pell equation x 2 − n y 2 = 1 {\displaystyle x^{2}-ny^{2}=1} with n = 8 {\displaystyle n=8} .
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Solutions (not necessarily optimal) have been computed for every N ≤ 10,000. [2] Solutions up to N = 20 are shown below. [2] The obvious square packing is optimal for 1, 4, 9, 16, 25, and 36 circles (the six smallest square numbers), but ceases to be optimal for larger squares from 49 onwards.