# Guess and Check, Work Backward

## Key Questions

• You should use the guess and check method when you do not know how to solve a problem.

The guess and check method includes:

1. make a logical guess
3. adjust your guess based on results of #2 until you are correct

Example:

There are 20 children in the kindergarten class. The children are a mix of 5 year olds and 6 year olds. The total age of the children equals 108 years. How many 5 year olds are there?

Guess & check method:

1. Let's guess that there are 10 five year olds.
2. If there are 10 five year olds, there must be 10 six year olds since there are 20 children in total. Their combined age is equal to (10 x 5) + (10 x 6), or 110 years.
3. Since 110 years is greater than 108 (the correct answer), our initial guess was incorrect. To get closer to the correct answer, we need to guess a higher number of five year olds (since five years is less than six years).
4. Let's now guess that there are 12 five year olds.
5. If there are 12 five year olds, there must be eight six year olds since there are 20 children in total. Their combined age is equal to (12 x 5) + (8 x 6), or 108 years. Therefore, the correct answer is 12 five year olds.

• I am not certain the following is a fully satisfactory answer to this question, but it explains the logic behind logical transformations from something we want to prove to an obviously true statement and back.

"Working backwards" is the typical final part of a proof when from a statement we want to prove, for instance ${x}^{2} - 4 x + 6 > 1$, we derive an obviously true statement using some invariant (equivalent) transformations like these:
${x}^{2} - 4 x + 6 = {\left(x - 2\right)}^{2} + 2$, which is greater or equal to 2 (since a square of any algebraic expression is greater or equal to $0$), which, in turn, is greater than $1$.

The problem with the above "proof" is that, if the initial statement was false, using seemingly correct transformations, we can come up to an obviously true statement. So, the fact that from our original statement we have derived the obviously true final statement does not necessarily prove that our initial statement was true.

But, if all transformations we made are not only "correct", but invariant (or equivalent), which, in short, means reversible, then after we have derived a true statement we can conclude:
since all transformations are invariant (that is reversible), from the final true statement we can derive the initial, and that is the actual proof.
This is actually the "working back" part of a proof.

For the example above the real proof is the following sequence:
${\left(x - 2\right)}^{2} \ge 0$ - add 2 to both sides -
${\left(x - 2\right)}^{2} + 2 \ge 2 > 1$ - open parenthesis -
${x}^{2} - 4 x + 4 + 2 > 1$ - simplify -
${x}^{2} - 4 x + 6 > 1$ - which is what we had to proof.