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Five Fives (Posted on 2005-09-18) Difficulty: 4 of 5
A mathematician who was exceedingly fond of the number five set to work trying to express as many consecutive integers using no numerals besides '5', and only up to five of them. She allowed herself to use any standard mathematical notation she knew, as long as it didn't contain any numerals. For example, she could use the symbol for 'square root', but not 'cube root' (because it contains a '3'). She determined that the highest consecutive integer she could express this way was 36. Her last few calculations were as follows:
  • 31 = 5*5 + 5 + (5/5)
  • 32 = 55*.5 + 5 - .5
  • 33 = (55 + 5) * .55
  • 34 = 5!/5 + 5/.5
  • 35 = (5 + (5+5)/5) * 5
  • 36 = 5*5 + 55/5
  • 37 = ?
Was she correct in thinking 36 was the highest consecutive integer she could express this way? Can you express 37 using only up to five 5's?

Note: The intention here is to find an exact expression, so rounding expressions like [] "greatest integer" are not allowed.
Note: Can you do it without using letters of any kind (x, log, lim, sum, etc.)?

See The Solution Submitted by Josh70679    
Rating: 4.4737 (19 votes)

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n!!!!!!!!! | Comment 268 of 393 |

Nice work Ady on introducing !n. I never heard of it before, but I like it. Moreover, upon my research I discovered there's a double factorial n!! We've been using n!! to express a factorial of a factorial (at least I have and it's used in josh's list), but it really should mean the product of all positive integers less than or equal to n that have the same parity (being even or odd) as n. (n!! = n * (n-2) * (n-4) * (n-6) )


n!!! would be n * (n-3) * (n-6) * (n-9)(through only the positive integers, of course)

from this we can get 10 with one five (5!!! = 5 * (5-3) = 10)


I don't know if you guys want to use this (personally, I think this would make it too easy), but we should at least start using a different notation for a factorial of a factorial. I suggest keeping it simple like (n!)! or for three !'s: ((n!)!)! I think that's the correct way anyway.


This is all kinda new to me, but Ady you seem like you know stuff If  I'm not on the right track here, set me straight.


* There also a superfactorial n$ (and some even more obscure), but that would be using a new symbol and it's not that useful anyway since the numbers get very large very fast. (n$ = n! * (n-1)! * (n-2)! ...* 1!)

Edited on October 14, 2005, 8:36 am

Edited on October 14, 2005, 10:18 am
  Posted by brad on 2005-10-14 08:36:14

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