Below are three groups of three numbers each. Combine the numbers in each group using the standard binary operations (addition, subtraction, multiplication, division, and exponentiation) so that each group yields the same number (there is one unique solution).

1. 1, 6, 11
2. 13, 20, 33
3. 20, 33, 40

For example, given:
15, 19, 24          11, 30, 36          20, 22, 36
you could make:
24÷(19-15)=6       (30+36)÷11=6         20+22-36=6

More than one solution...but only one unique value where the mathematical operations will yield the same number.
The unique number yielded from each of the three sets is 66.

There are 14 combinations where {1,6,11} and the standard binary operations will equal 66. Due to the commutative property of multiplication, of the 14, there are 6 combinations where both binary operators is multiplication:
1*6*11 = 1*11*6 = 6*1*11 = 6*11*1 = 11*1*6 = 11*6*1 = 66
Due to the associative property of multiplcation and division, and the commutative property of multiplcation, of the 14, there are 4 combinations where there is one each of the binary operators of multiplcation and division:
6*11/1 = 6/1*11 = 11*6/1 = 11/1*6 = 66
Due to the order of operations and the commutative property of multiplication, of the 14, there are 4 combinations where the binary operators are multiplication and exponentiation:
6^1*11 = 6*11^1 = 11*6^1 = 11^1*6

There are 4 combinations where {20,33,40} and the standard binary operations will equal 66. Due to the associative property of multiplcation and division, and the commutative property of multiplication, there are 4 combinations where there is one each of the binary operators of multiplcation and division:
33/20*40 = 33*40/20 = 40/20*33 = 40/20*33 = 11/1*6 = 66

There are 6 combinations where {13,20,33} and the standard binary operations will equal 66. Due to the commutative property of addition, there are 6 combinations where the two binary operators are both addition:
13+20+33 = 13+33+20 = 20+13+33 = 20+33+13 = 33+13+20 = 33+20+13 = 66

Posted by Dej Mar on 2012-09-23 06:42:17