Solution for the sum 144 using 6 sixes?

Answer 1

The answer to most of these types of brain teasers is to know that the number is not limited to the number six, but caps the number of characters used (as in the equation will only show the numeral 6 six times).

The quickest way to solve the teaser is trial and error (perfectly reasonable math or scientific process in some instances.)

Numbers containing only 6's less than 144 are 66 and 6. Using the larger of the two numbers:

144-66 = 78 (four 6's left) --> 78-66 = 12 (two 6's left)

2*6 = 12 so a solution is 66+66+6+6 = 144

**The Mathematical and long way to prove the solution**

Since the problem makes clear that the addition operation is our only available math tool we can determine how many sixes we need to add together to get 144: 144/6 = 24. Numbers only showing 6 less than 144 are 66 and 6 which can also be written as 6*11 = 66 and 6*1 = 6.

Combining these mathematical statements provides the following equation:

6*24 = 6(x*11+y) --> 24 = 11*x + y

(where x and y are the number of 66's and 6's, respectively, in the equation we are building.)

We also know that 2*x+y = 6 (Two 6's in 66 for every x and one 6 in 6 for every y. The total 6's in the equation are limited to six.)

Solving the equations 11*x+y = 24 and 2*x+y = 6 using Gaussian Elimination:

11*x+1*y = 24

-(2*x+1*y) = -(6)

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9*x+0*y = 18 --> x = 18/9 = 2 (There are two 66's in the equation.)

2*2+y = 6 --> y = 6-4 = 2 (There are two 6's in the equation.)

It is proven that the final and only solution is 66+66+6+6 = 144