Reasoning · Grade 3-2 Cryptarithms and Hidden-Digit Puzzles
Problem
B × B must end in B, leaving 0, 1, 5, 6.
A multiplication fact only repeats its own ones digit for 0, 1, 5, and 6 — something a Grade 3 student can see just from the times tables.
3.OA.C.7Look For A PatternA digit can only reappear as the ones digit of its own multiple for 0, 1, 5 and 6, so B is one of those four.
The ones digit of a product depends only on the ones digits multiplied, because every higher place carries whole tens upward.
Trying all ten digits against that one test throws out six of them at once, leaving a list short enough to check by hand.
So the answer to part 1 is 0, 1, 5, 6.
Listing the survivors keeps the search small — only four numbers to think about instead of ten.
3.OA.C.7Make A Systematic ListB leads a number so 0 is out, and 1 leaves a two-digit partial product, so only 5 and 6 remain.
A leading digit is never 0, and multiplying by 1 cannot make a number longer — both are quick sanity checks.
3.OA.A.4Eliminate PossibilitiesTesting B = 5, 75 × 5 = 375 fits CAB exactly.
Guessing the natural candidate 75 and checking instantly confirms it — the digits line up exactly.
3.OA.C.7Guess And CheckThe second partial 75 × 7 = 525 gives D = 2.
The tens partial product is just another single-digit times AB, a basic Grade 3 fact.
3.OA.C.7Guess And CheckThe whole product 75 × 75 = 5625 gives E = 6.
Adding the two partial products (one shifted left) is exactly how column multiplication builds the final answer.
3.NBT.A.2Guess And CheckNo case with B = 6 works, so the answer is unique.
Checking the only other survivor confirms the solution is unique — no other digit choice works.
3.OA.A.4Make A Systematic ListOnly the digits 0, 1, 5, and 6 copy themselves when you multiply them by themselves — that one trick cracks the whole puzzle with just your Grade 3 times tables!