Technology

Can You Solve This Number Puzzle?

February 24, 2016 | Elizabeth Knowles

A+B+C = AxBxC. What non-zero whole numbers are a, b, and c?

Give it a shot!

You might have heard that it takes two equations to solve for two variables, three equations to solve for three variables, etc. In this case we only have one equation with three variables, but it is still easy to find a solution:

a=1

b=2

c=3

1+2+3=6

1x2x3=6

That’s all there is to it! ...or is it?

Since 1-2-3 was probably just an educated guess, you might wonder whether a second solution exists. Let's try to find one:

SEE ALSO: Are Powerball Drawings and ‘Quick Pick’ Numbers Really Random?

Remember that ≤ means “smaller or equal to,” < means “smaller than,” and ≥ means “greater or equal to.”

Let’s assume that a ≤ b ≤ c. Since a, b and c are any three numbers, it makes sense for us to put them in order of size.

From that assumption, we can say that a+b+c ≤ 3c.

Since c > 0, if 3<ab (we’re just assuming here), then we can multiply both sides by c and get 3c<abc.

So a+b+c ≤ 3c < abc.

Since we know that a+b+c = abc, these inequalities don’t work — there is no possible value for 3c — which tells us that our assumption was wrong and that ab ≤ 3.

There are only three pairs of possible numbers here for a and b: (1,1), (1,2) and (1,3). We have to test out each pair individually.

(1,1):

1+1+c=1*1*c

2+c=c

2=0

Nope, that one doesn’t work.

(1,2):

This is the solution we already have.

(1,3):

1+3+c=1*3*c

4+c=3c

4=2c

c=2

This is the same solution as the previous pair that we just tried, except that the values for b and c were switched. Therefore, the only solution to the equation is 1,2,3.

If you liked this brain teaser, try this one next: The Sleeping Beauty Puzzle Has Two Contradictory Correct Answers