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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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Some Thoughts Can you express 1037 with only up to five 5's? | Comment 390 of 391 |

In the following, a number preceded by a raised dagger () or daggers indicates a reference. The number of daggers indicate the number of 5's used to form the number. The first list is provided simply for reference, some with a note indicating the type of symbol notation used. The second list is the list of numbers 1001 to 1036. It is possible to generate higher sequences, yet I have not yet discovered how to form 1037.

Take note that a limit is now imposed that restricts the number of any distinct symbols to 5, just as the limit has been for the number of fives. Though the following conforms to those restrictions, it is not readily seen in the following.

  1/2   .5
  5/9   .5... --- non-terminating decimal
    1   5 / 5
    2   30 / 15
    3   15 / 5
    4   120 / 30
    6   30 / 5
    8   120 / 15
    9   5 / .5... --- non-terminating decimal {.n...}
   10   5!!! --- multifactorial, 3rd degree {n!!!}
   12   (3)$ --- superfactorial {n$}
   15   5!! --- double factorial {n!!}
   30   5# --- primorial {n#}
   44   !5 --- subfactorial {!n}
  120   5! --- factorial {n!}
  900   30 * 30
  945   (9)!!
15120   <5>5 --- rising factorial {<x>n}
===============================================
 
1001   945 + 44 + 12
1002   1008 - 6
1003   1008 - 5
1004   1008 - 4
1005   1008 - 3
1006   1008 - 2
1007   1008 - 1
1008   15120 / 15
1009   1008 + 1
1010   1008 + 2
1011   1008 + 3
1012   1008 + 4
1013   1008 + 5
1014   1008 + 6
1015   1020 - 5
1016   1020 - 4
1017   1020 - 3
1018   1020 - 2
1019   1024 - 5
1020   900 + 120
1021   1020 + 1
1022   1020 + 2
1023   1020 + 3
1024   (4)5
1025   1024 + 1
1026   1024 + 2
1027   1024 + 3
1028   1024 + 4
1029   1024 + 5
1030   1020 + 10
1031   (120 * 9) - 44 - 5
1032   (120 + 9) * 8
1033   1008 + 30 - 5
1034   1020 + 44 - 30
1035   1020 + 15
1036   1024 + 12
1037   ?

 

Edited on December 16, 2012, 2:37 am
  Posted by Dej Mar on 2012-12-16 02:06:11

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