All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
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.)
A different one (to Josh) | Comment 6 of 391 |

First, OOO, exclude your third answer, because log is not allowed.

Second, I didn´t undertood Dimmeh´s solution.

To Josh: searching a different way, I elaborate this one, and the support you´ll find in the link that I´m posting

http://mathforum.org/kb/message.jspa?messageID=3782901&tstart=0

 

"However D.E. Knuth has suggested (for the triangular numbers) the name "termial function" and defines it as n? = 1+2+3+...+n = n*(n+1)/2. And he evaluates n? for not only natural numbers, but also rationals (1/2? = 3/8)."

 

And, using a dot over the 5 to indicate "repeating decimal", as allowed in all problems of this kind, my solution is:

 

                    .

       (5 * 5 / .5) - (5! / 5?) = 45 - 8 = 37


  Posted by pcbouhid on 2005-09-18 17:37:02
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (5)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information