Four 1's puzzle

Uncwilly · 2007-09-21, 22:43

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 1

Addition: 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 2

Factorial: 11! = 39916800
Exponential: 11^11 = 285311670611
Square-root: √11 = 3.3166247903553998491149327366707

Class 3

Absolute 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

Mini-Geek · 2007-09-21, 23:35

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.

retina · 2007-09-22, 01:40

10 = 11 - 1 / 1 (2 points)
12 = 11 + 1 / 1 (2 points)
100 = 11 / .11 (1.5 points)

Kevin · 2007-09-22, 03:27

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.

Mr. P-1 · 2007-09-22, 04:12

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)

retina · 2007-09-22, 05:23

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?

retina · 2007-09-22, 05:26

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)

Mr. P-1 · 2007-09-23, 04:55

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?

retina · 2007-09-23, 06:10

-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.

retina · 2007-09-24, 04:02

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?

Kevin · 2007-09-24, 06:00

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.

2007-09-21, 22:43	#1
Uncwilly 6809 > 6502 """"""""""""""""""" Aug 2003 101×103 Posts 2²·2,371 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 1 Addition: 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 2 Factorial: 11! = 39916800 Exponential: 11^11 = 285311670611 Square-root: √11 = 3.3166247903553998491149327366707 Class 3 Absolute 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

2007-09-22, 05:26	#7
retina Undefined "The unspeakable one" Jun 2006 My evil lair 29·211 Posts	5 = 11/(.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*

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2007-09-21, 23:35	#2
Mini-Geek Account Deleted "Tim Sorbera" Aug 2006 San Antonio, TX USA 10253₈ 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 17E7₁₆ 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 7·167 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)

2007-09-23, 06:10	#9
retina Undefined "The unspeakable one" Jun 2006 My evil lair 29×211 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 29·211 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?