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.)?

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