Letter Substitution
The setup
In these puzzles, single digits are replaced by letters. At first glance, these puzzles seem to be the same as the ‘Missing Number’ puzzles from earlier in this Stage. However, the use of letters provides more interesting opportunities for problem solving. If your child is comfortable with the Missing Number puzzles, you should transition to these puzzles.
The use of letters in these puzzles follows three rules:
- A given letter is always the same digit from 0 to 9
- The leftmost digit of a number is never 0
- Different letters must be different digits
How to create
Take an ordinary addition or subtraction problem and replace one or more of the digits. Use the same letter when replacing the same digit. In this example, 6 is replaced by ‘A’ in both places.
Special puzzles
The circumstances of this type of puzzle allow for the creation of interesting problem-solving challenges. These take a bit of designing, but the result is some fun puzzles.
Notice that the values of the letters do not carry over from puzzle to puzzle. The ‘B,’ which has value 1 in this first puzzle, has value 4 in the second.
Bonus Material
Introduction
Once your child becomes comfortable with the Missing Number puzzles from a few pages earlier in this chapter, they can start playing with these puzzles. In these, one or more of the digits are replaced by letters. The three rules for letters are:
- A given letter is always the same digit
- The leftmost digit of a number is never 0
- Different letters must be different digits
Create these puzzles by taking an addition or subtraction problem and replacing one or more of the digits. The puzzles can also be created to make interesting problem-solving challenges for your child. Note that the values of the letters do not carry over from puzzle to puzzle.
Examples
This first example illustrates how you can take a standard addition or subtraction problem and make a letter substitution puzzle out of it. The first version replaced all the 6’s with A’s, and the second version went on to replace the 2’s with B’s.
The rest of these examples are carefully constructed to allow solving using properties of the particular situation. One property to note is that when you add two numbers, the carry into the next column is always either 0 or 1. So, for example, in the problem A + A = C4, C must be 1 because it is not allowed to be 0.
Helping your child
Puzzles are meant to be challenging and to take time, so please don’t ruin the fun by telling your child how to do them. These puzzles are chosen so that you can create them easily and then have fun solving them together.
If your child gets stuck on a puzzle, you have several options. You can, of course, give very small hints, if you can think of things that won’t give away the puzzle. You can suggest looking at smaller or simpler versions of the puzzle. Encourage your child to be bold in their ideas, even if sometimes they lead to dead ends. We all learn a lot from our mistakes and dead ends! Let your child know that it is perfectly okay not to solve a puzzle on the first (or second or third) try, and that useful ideas may occur to them if they leave the puzzle alone for a day or two.
These puzzles are meant to be fun and to teach problem solving. One of the greatest mathematical pleasures is that AHA moment, after many false starts and much wrestling with a problem, when the answer is finally discovered – be sure to let your child experience that feeling of discovery as many times as you can!