Puzzles
Odd sums
What is \(\frac{1+3}{5+7}\)?
What is \(\frac{1+3+5}{7+9+11}\)?
What is \(\frac{1+3+5+7}{9+11+13+15}\)?
What is \(\frac{1+3+5+7+9}{11+13+15+17+19}\)?
What is \(\frac{\mathrm{sum\ of\ the\ first\ }n\mathrm{\ odd\ numbers}}{\mathrm{sum\ of\ the\ next\ }n\mathrm{\ odd\ numbers}}\)?
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They are all equal to one third.
The sum of the first \(n\) odd numbers is \(n^2\) (this can be proved by induction). This means that:
$$\frac{\mathrm{sum\ of\ the\ first\ }n\mathrm{\ odd\ numbers}}{\mathrm{sum\ of\ the\ next\ }n\mathrm{\ odd\ numbers}}=\frac{n^2}{(2n)^2-n^2}\\
=\frac{n^2}{3n^2}=\frac{1}{3}$$
Extension
What is \(\frac{\mathrm{sum\ of\ the\ first\ }n\mathrm{\ odd\ numbers}}{\mathrm{sum\ of\ the\ first\ }n\mathrm{\ even\ numbers}}\)?
x to the power of x
If \(x^{x^{x^{x^{...}}}}\) [\(x\) to the power of (\(x\) to the power of (\(x\) to the power of (\(x\) to the power of ...))) with an infinite number of \(x\)s] is equal to 2, what is the value of \(x\)?
Twenty
How many three digit integers are there for which the product of the digits is 20?
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There are two ways to make 20 by multiplying three digits: \(2\times 2\times 5\) and \(1\times 4\times 5\). Listing all the possible orderings of these, we have:
145
154
415
451
514
541
225
252
522
Therefore, there are 9 different three digit numbers where the product of the digits is 20.
Extension
How many 4 digit numbers are there where the product of the digits is 20?
5 digit?
\(n\) digit?
8! minutes
How many weeks are there in 8! (\(8\times 7\times 6\times 5\times 4\times 3\times 2\times 1\)) minutes?
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$$8\times 7\times 6\times 5\times 4\times 3\times 2\times 1 \mbox{ minutes}$$
$$= 8\times 7\times 4\times 3\times 1 \mbox{ hours}$$
$$= 7\times 4\times 1 \mbox{ days}$$
$$= 4\times 1 \mbox{ weeks}$$
$$= 4 \mbox{ weeks}$$
Extension
8 is the smallest number \(n\) such that \(n!\) minutes is a whole number of weeks.
What is the smallest number \(m\) such that \(m!\) seconds is a whole number of weeks?
