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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)

Comments: ( Back to comment list | You must be logged in to post comments.)
re: 12 and 317 | Comment 237 of 391 |
(In reply to 12 and 317 by brad)

nice work on 317. 

Nice work getting 12 also, though there's been some question as to whether % counts as 'standard mathematical notation'.  There's a way to get 292 using that expression for 144, but I opted to wait to see if we could find an expression without '%'.  As far as i'm concerned, go ahead and use % if you have to, but try to get around it if possible.

BTW, would you like me to send you the full list I've accumulated?


  Posted by Josh70679 on 2005-10-10 14:00:49
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