Reasoning · Grade 3-2 Cryptarithms and Hidden-Digit Puzzles

Problem

Recover Erased Multiplication Digits

Digits are hidden in a column multiplication whose top number starts with 2. The first partial product is two digits ending in 1; the second ends in 2. The final product is four digits with 0 in the tens place. Recover every hidden digit.
Your answer
How to solve
Strategy Guess and Check — Break the column into separate pieces (the ones-digit partial product, the tens-digit partial product, the final sum) and use the ones-digit clues to limit each hidden digit to a few values, then guess and check the survivors until all three lines agree.
1STEP 1

Use the first partial product to limit X and u

Ending in 1 needs a pair like 3 × 7.

X × u ≡ 1 (mod 10)
2STEP 2

Pin down X = 7 and u = 3

Staying two digits leaves only 27 × 3 = 81.

27 × 3 = 81
3STEP 3

Find the multiplier's tens digit t

Only 27 × 6 = 162 ends in 2.

27 × 6 = 162
4STEP 4

Compute the final product and check the tens digit

So the multiplier is 63 and the product 1701.

27 × 63 = 81 + 1620 = 1701
5STEP 5

Fill every box

1701 has 0 in the tens, matching the clue.

27 ; × 63 ; 81 ; 162 ; 1701
Answer
27 × 63 = 1701 (partials 81 and 162)
Re-check each clue: first partial product 81 ends in 1 (yes, two digits); second partial product 162 ends in 2 (yes); final product 1701 is four digits with tens digit 0 (yes). And 27 × 63 = 1701 directly, so everything is consistent.
Takeaway

The ones digit of a product is a giant clue — once you spot that 7 × 3 ends in 1, the whole hidden multiplication unlocks with Grade 3 times tables!

  • Use the first partial product to limit X and u
  • Pin down X = 7 and u = 3
  • Find the multiplier's tens digit t
  • Compute the final product and check the tens digit
  • Fill every box