Puzzles
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?
Show answer & extension
Hide answer & extension
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?
Show answer & extension
Hide answer & extension
$$8\times 7\times 6\times 5\times 4\times 3\times 2\times 1 \mbox{ minutes}$$
$$= 8\times 7\times 4\times 3\times 1 \mbox{ hours}$$
$$= 7\times 4\times 1 \mbox{ days}$$
$$= 4\times 1 \mbox{ weeks}$$
$$= 4 \mbox{ weeks}$$
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?
