Problem
Cryptarithm: letters stand for digits
The four-digit number 45AB times 6 gives the five-digit number 27BBB. Every B stands for the same digit. A and B stand for different digits. Find the digit each letter is.
Reasoning · Grade 4-1 Arithmetic Puzzles
Your answer
Watch out Even when the arithmetic works, A equal to B is not allowed.
How to solve
Strategy Eliminate Possibilities — B has only ten possible values, so this really is a small finite candidate set. Two different clues cut that set down fast: the ones digit of the product must come back as B, and the whole product must start with 2 and 7, which pins its size. After those two cuts only three values of B survive, and each can be tested by a single short multiplication.
1STEP 1
Use the ones digit to narrow B
The ones digit forces B to be even.
6 × 0 = 0, 6 × 2 = 12, 6 × 4 = 24, 6 × 6 = 36, 6 × 8 = 48
The ones digit of a product depends only on the ones digits being multiplied, so ten quick facts from the six times table test all ten candidates at once.
4.NBT.B.5Make A Systematic ListThe ones digit of the product depends only on the ones digits multiplied, so one row of the six times table tests all ten candidates.
Why?
Every place above the ones only ever contributes whole tens to a product, so it can never change what lands in the ones place.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
Why?
Every digit whose product ends in the wrong figure is struck out at once, so the list shrinks before any real work starts.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
2STEP 2
Use the size of the number to narrow B further
The product cannot pass 27594, so B is at most 5.
6 × 4500 = 27000, 6 × 4599 = 27594
Squeezing an unknown number between two round numbers you can multiply easily gives a size range for the answer before you know any of its hidden digits.
4.NBT.B.5Eliminate Possibilities3STEP 3
Cut the candidate list down to three
That leaves only 0, 2 and 4.
27BBB ≤ 27594 → B ≤ 5
Comparing two five-digit numbers place by place from the left is enough to bound a hidden digit, and combining two separate clues shrinks ten candidates to three.
4.NBT.A.2Eliminate Possibilities4STEP 4
Test the three survivors
Testing each, only B = 4 works.
27444 - 27024 = 420, 420 ÷ 60 = 7
Knowing that each step of A adds a fixed 60 turns ten guesses per case into one subtraction, and a gap that is not a multiple of 60 rules the whole case out at once.
4.NBT.B.5Guess And Check5STEP 5
Throw out the case where the letters would clash
Then A = 7, and the two digits differ.
In a letter puzzle the naming rule is a real condition, not decoration, and here it is the only thing that rules out the second arithmetic solution.
4.NBT.A.2Eliminate Possibilities6STEP 6
Check the finished multiplication
Checking gives 4574 × 6 = 27444.
4574 × 6 = 27444
Redoing the multiplication column by column confirms that every digit of the puzzle really lines up, not just the ones that were used to find the answer.
4.NBT.B.5Guess And CheckAnswer
A = 7, B = 4
4574 × 6 = 27444
The finished statement 4574 times 6 equals 27444 checks out place by place, and its size is sensible: 4574 is a little over 4500, and 6 times 4500 is 27000, so a product just over 27000 is exactly what to expect. The letters obey the rules too, since A is 7, B is 4 and the two are different digits.
Takeaway
Two easy clues, what the ones digit must be and how big the answer can be, shrink ten guesses down to one.
- Use the ones digit to narrow B
- Use the size of the number to narrow B further
- Cut the candidate list down to three
- Test the three survivors
- Throw out the case where the letters would clash
- Check the finished multiplication