Problem
Solve a cryptarithm for each letter
In the addition SO + SO = TOO, each letter is a single digit. The same letter is always the same digit, and different letters are different digits. We must find which digit each of S, O, and T stands for.
Reasoning · Grade 2-1 Cryptarithms and Magic Squares
Your answer
Watch out S + S = 0 also ends in 0, but S leads the number so it cannot be 0 — which forces S + S = 10.
How to solve
Strategy Work Backwards — We are given the finished addition (the end state) and must recover the digits that produced it. Working column by column from the ones place lets the result digits pin down each letter, and a quick check confirms the only possibility.
1STEP 1
Read the ones column
Stacked up, the ones column is O + O and the answer also ends in O — so O + O must end in O itself.
O + O → O
Lining the numbers up by place value is exactly how Grade 2 students add two-digit numbers, one column at a time.
2.NBT.A.1Work Backwards2STEP 2
Find O from the ones column
The only digit that survives doubling in the ones place is O = 0, and nothing carries.
0 + 0 = 0
A child can just try doubles 0+0, 1+1, 2+2, ... and notice only 0 lands back on itself.
2.OA.B.2Guess And CheckIn the ones column O plus O has to end in O again, and 0 is the only digit that does that.
Why?
Adding a number to itself and getting the same last digit back means the second copy added nothing at all to that place.
Why?
Zero is the one amount you can add without changing the number, so only zero can leave a digit exactly as it was.
🧱Additive identity (zero)Adding zero adds nothing, so the number stays exactly the same.
Why?
The ones column can be judged on its own because each column of a written sum handles one size of bundle and passes only whole bundles upward.
🧱Place-value groupingTen ones make one ten and ten tens make one hundred — each place bundles ten of the last.
3STEP 3
Read the tens column
With no carry the tens column is S + S, ending in 0 — so it is 0, 10, or 20.
S + S → 0
Again this is column addition: the bottom of the tens column has to match the answer's tens digit.
2.NBT.B.5Work Backwards4STEP 4
Find S and the carry that makes T
S leads the number so it isn't 0, giving S + S = 10 — S = 5 and the carried 1 becomes T.
5 + 5 = 10 → S = 5, T = 1
Knowing the doubles, only 5 + 5 makes a ten, and the leftover ten becomes the new hundreds digit.
2.NBT.B.5Guess And Check5STEP 5
Check all letters are different
5, 0, 1 are all different, so the cryptarithm rule holds.
S = 5, O = 0, T = 1
Listing the found digits side by side makes it easy to confirm none repeat.
2.NBT.A.3Make A Systematic ListAnswer
S = 5, O = 0, T = 1 (50 + 50 = 100)
50 + 50 = 100
Substitute back: SO = 50, so 50 + 50 = 100 = TOO with T = 1, O = 0, O = 0. The sum of two two-digit numbers giving a three-digit number is reasonable, and every column matches.
Takeaway
This only needs the Grade 2 column-by-column addition you already know -- start at the ones place and the letters fall right out!
- Read the ones column
- Find O from the ones column
- Read the tens column
- Find S and the carry that makes T
- Check all letters are different