mersenneforum.org Four 1's puzzle
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 2007-09-21, 22:43 #1 Uncwilly 6809 > 6502     """"""""""""""""""" Aug 2003 101×103 Posts 100101000001002 Posts Four 1's puzzle Figured I would start this one. Like the 4 4's. Using four 1's and standard mathematical operations formulate equations with the answer being each of the numbers from 0 to 100 and on. Every calculation MUST use exactly all four 1's and no other digits. The allowable operations are: Class 1Addition: 1 + 1 = 2 Subtraction: 1 – 1 = 0 Multiplication: 1 * 1 = 1 Division: 1 / 1 = 1 Parenthesis: (1 + 1) / 1 = 2 Concatenation: 111 Decimal Point: .1 Class 2Factorial: 11! = 39916800 Exponential: 11^11 = 285311670611 Square-root: √11 = 3.3166247903553998491149327366707 Class 3Absolute value: |-1| = 1 gamma: ? Solution scoring: 100 points per class 3 operator 10 points per class 2 operator 1 point per class 1 operator (parenthesis per pair), except the following: 1/2 point per decimal 0 points for concatenation The goal is to get as low as score solution as possible. First 3 to start: 0 = 11 - 11 1 = 11 / 11 2 = 1/1 + 1/1 Last fiddled with by Uncwilly on 2007-09-21 at 22:47
 2007-09-21, 23:35 #2 Mini-Geek Account Deleted     "Tim Sorbera" Aug 2006 San Antonio, TX USA 17·251 Posts I'm putting the score for it next to the number. A few in order: 3, 3 = 1 * 1 + 1 + 1 3, 4 = 1 + 1 + 1 + 1 And some easy ones out of order: 2, 9 = 11 - 1 - 1 3, 10 = 11 - (1 * 1) 2, 11 = 11 * 1 * 1 3, 12 = 11 + (1 * 1) 2, 13 = 11 + 1 + 1 1, 22 = 11 + 11 1, 110 = 111 - 1 1, 111 = 111 * 1 1, 112 = 111 + 1 1, 121 = 11 * 11 0, 1111 = 1111 10, 285311670611 = 11 ^ 11 I'm not so sure that very many numbers can be made with four 1's, just because of how it's so low and many operations don't, or barely, change the number you start with.
 2007-09-22, 01:40 #3 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 41×149 Posts 10 = 11 - 1 / 1 (2 points) 12 = 11 + 1 / 1 (2 points) 100 = 11 / .11 (1.5 points)
 2007-09-22, 03:27 #4 Kevin     Aug 2002 Ann Arbor, MI 1B116 Posts I agree that we need to add at least one degree of fudging to get everything as as combination of some fixed amount of 1's. I was thinking different number bases, but sadly $b=11_b-(1_b*1_b)$ for all b>1.
 2007-09-22, 04:12 #5 Mr. P-1     Jun 2003 100100100012 Posts 5 = (1 + 1 * 1)/.1 (4.5 points) 6 = (1 + 1 + 1 * 1)! (14 points) 7 = (1 + 1 + 1)! + 1 (14 points) 8 = 1/.1 - 1 - 1 (3.5 points)
2007-09-22, 05:23   #6
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

41×149 Posts

Quote:
 Originally Posted by Mr. P-1 5 = (1 + 1 * 1)/.1 (4.5 points)
Really? I get 20 for that sum. What did I do wrong?

 2007-09-22, 05:26 #7 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 41·149 Posts 5 = 1*1/(.1+.1) (5 points) 10 = 11/1.1 (1.5 points) 20 = 1/.1+1/.1 (4 points) 24 = (1+1+1+1)! (14 points) 30 = (1+1+1)/.1 (4.5 points) 55 = 11/(.1+.1) (4 points) 60 = (1+1+1)!/.1 (14.5 points) 109 = 11/.1-1 (2.5 points) 1110 = 111/.1 (1.5 points) $\infty$ = 11/(1-1) (3 points) Last fiddled with by retina on 2007-09-22 at 06:13
2007-09-23, 04:55   #8
Mr. P-1

Jun 2003

22218 Posts

Quote:
 Originally Posted by retina Really? I get 20 for that sum. What did I do wrong?
You didn't. I did. (Sigh)

I also had (1 + 1 + 1)! -1 for five, but your solution is even better.

Has anyone been able to get 14?

 2007-09-23, 06:10 #9 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 10111110111012 Posts -1 = 1/.1 - 11 (2.5 points) 21 = 1/.1 + 11 (2.5 points) Uncwilly, this puzzle is really hard. I hope you plan on eventually posting the solutions, because I think everyone is stuck. There must be something fundamental that we all seem to have missed so far. So many gaps in the range 0-100. Perhaps a small hint is in order.
 2007-09-24, 04:02 #10 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 137358 Posts 99 = 1/.1/.1-1 (4 points) 101 = 1/.1/.1+1 (4 points) 200 = (1+1)/.1/.1 (5 points) 719 = (1+1+1)!!-1 (24 points) 720 = (1+1+1)!!*1 (24 points) 721 = (1+1+1)!!+1 (24 points) 1000 = 1/.1/.1/.1 (4.5 points) 1100 = 11/.1/.1 (3 points) I can't see how √, || or gamma can be useful. Is that a clue as to how to generate the missing values in the 0-100 range?
2007-09-24, 06:00   #11
Kevin

Aug 2002
Ann Arbor, MI

43310 Posts

Quote:
 Originally Posted by retina 99 = 1/.1/.1-1 (4 points) 101 = 1/.1/.1+1 (4 points) 200 = (1+1)/.1/.1 (5 points) 719 = (1+1+1)!!-1 (24 points) 720 = (1+1+1)!!*1 (24 points) 721 = (1+1+1)!!+1 (24 points) 1000 = 1/.1/.1/.1 (4.5 points) 1100 = 11/.1/.1 (3 points)
Just as a minor point, you probably want to write it at ((1+1+1)!)!. Usually !! is taken to mean n(n-2)(n-4)... (http://mathworld.wolfram.com/DoubleFactorial.html).

Also, gamma can be useful if you wanted to get (n-1)! instead of n! without wasting a 1.

Last fiddled with by Kevin on 2007-09-24 at 06:12

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