Concept 5 of 7
Cryptarithm Puzzles: Find the Missing Digit
Today's big idea
Start at the units column, since it is the only one with no carry coming in.
A leading letter never stands for zero, or the number would lose a digit.
How it works
👀 What's going on?
A sum arrives with letters standing where digits should be, and it looks as though there is nowhere to start. There always is a place to start, and it is the right-hand end.
💡 Here's the trick
Look at the sum above. Work the units column first, because it is the only column with no carry arriving into it. Each column you finish hands you one digit and possibly a carry for the next.
🪄 How to do it
- 1.Start at the units column of the sum above.
- 2.Pin the letter there, and note any carry it produces.
- 3.Move one column left, carrying that amount with you.
✍️ Try it
In 5A plus 26 equals 83, what is A? Then: in 4B plus B4 equals 88, what is B?
In the first, A plus 6 must end in 3, so A is 7, carrying 1. The tens then read 5 plus 2 plus 1. In the second, 44 plus 11 times B equals 88, so B is 4.
⚠️ Watch out
You might be thinking a letter can stand for any digit at all, zero included. The leading letter of a number never stands for zero, because the number would then have fewer digits than it shows.
🌟 Why this matters
Reading a calculation one column at a time, watching the carry, is the same habit that catches an error in a bill.
Exam tip: Write the carry above the next column as soon as you find it, before you forget it exists.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“In the cryptarithm AB + 27 = 65, each letter is a digit. Find A and B.”
- 1
Units column first
B + 7 ends in 5, so B = 8 (8 + 7 = 15, carry 1).
- 2
Tens column, add the carry
A + 2 + 1 = 6, so A = 3.
- 3
Check the whole sum
38 + 27 = 65 ✓
- 4
Answer ✓
A = 3, B = 8
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Same letter = same digit. Start at the units column, work out the carry, then finish.
CRYPTARITHM · ROUND 1 OF 3
2 A + 4 8 = 7 3
Units: A + 8 ends in 3, so A = ?
Quick check
✅ VERIFICATION
In the sum A5 + 27 = 82, what digit does A stand for?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Estimation of Products
Close-enough beats exact when speed matters.