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What comes next in this series?
0, 1, 2, 720!, ?
In math, the exclamation mark is a factorial, which is the product of every whole number from 1 to n.
For example, five factorial is written 5! = 1x2x3x4x5 = 120. By definition 0! = 1.
So 720! = 1x2x3x4x...718x719x720, which is a very large number.
0, 1, 2, 720!, ?
In math, the exclamation mark is a factorial, which is the product of every whole number from 1 to n.
For example, five factorial is written 5! = 1x2x3x4x5 = 120. By definition 0! = 1.
So 720! = 1x2x3x4x...718x719x720, which is a very large number.
Hint
Look at the first few factorials:0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
Answer
The crucial insight is that 6! = 720, so you can rewrite the series as:0, 1, 2, (6!)!, ?
Then note that 3! = 6, so you can rewrite the series again as:
0, 1, 2, ((3!)!)!, ?
You might think the series is:
0!!!, 1!!!, 2!!!, 3!!!, ...
However, 0!!! = 1, so the series must be:
0, 1!, 2!!, 3!!!, ...
Thus, the next term is 4!!!!.
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Comments
Good one, Marc. Pretty tricky. I was quite pleased being able to solve it!!!!
Really outstanding. I went down several rabbit holes. First I looked for interpretations, 720! is the number of ways to arrange minutes in twelve hours. Two the number of 12 hour periods (a.m./p.m.) per day, 1 the number... uh...
Then I looked for patterns that would just fit 0,1,2,x,... for any x. Nope.
Then I looked for solutions where the next number was based on the previous one or two, like a Fibonacci (sp?) variant. Nope.
Then, finally, I worked down to it. Tough since the 1 and two could be followed by any number of factorials.
Well done! It's always exciting to see new math teasers! Apologies to those for whom this will not be their cup of tea.
Then I looked for patterns that would just fit 0,1,2,x,... for any x. Nope.
Then I looked for solutions where the next number was based on the previous one or two, like a Fibonacci (sp?) variant. Nope.
Then, finally, I worked down to it. Tough since the 1 and two could be followed by any number of factorials.
Well done! It's always exciting to see new math teasers! Apologies to those for whom this will not be their cup of tea.
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