Problem
Finding the hidden rule
Each group has four outer circles and one in the middle. One hidden rule turns the outer numbers into the middle one in every group. Find the rule from the three filled groups and complete the fourth.
Reasoning · Grade 3-2 Multiplication Patterns
Your answer
Watch out A rule that fits the first group is easy to invent — it must fit all three.
How to solve
Strategy Look for a Pattern — I have three worked groups, so I can guess a combining rule (a sum, a product, or a mix of the two diagonals) and check it against all three. Once one rule fits every solved group, I apply it to Group 4.
1STEP 1
Test a rule on Group 1
Adding gives 14, short of 26 — addition alone fails.
(3×2)+(4×5) = 6+20 = 26
Since plain addition is far too small, multiplying the pairs first is the natural next guess to reach a bigger number like 26.
3.OA.D.9Guess And Check2STEP 2
Confirm the rule on Groups 2 and 3
Multiplying top pair and bottom pair then adding fits all three.
(2×6)+(3×7)=33, (4×6)+(8×2)=40
A rule that correctly reproduces all three known centers is almost certainly the hidden pattern.
3.OA.D.9Look For A PatternA rule that reproduces all three known centres can be trusted on the fourth group.
Why?
A single group where the rule gives the wrong centre would prove it false, so surviving every known group is real evidence.
🧱One counterexample refutes an always-claimA claim about every case dies the moment you exhibit one case where it fails, and exhibiting is easier than arguing.
Why?
The rule multiplies each top and bottom pair, and a pairing that works the same way in every group is what makes it a rule rather than a coincidence.
🧱Multiplication as equal groupsa times b means a equal groups of b — it is just repeated adding of the same amount.
3STEP 3
Apply the rule to Group 4
Applied to the fourth: 7 × 5 + 6 × 4 = 59.
(7×5)+(6×4) = 35+24 = 59
Multiplying each top/bottom pair and adding uses only basic multiplication facts and a single addition, well within Grade 3.
3.OA.C.7Look For A PatternAnswer
59
7 × 5 + 6 × 4 = 59
The center number is roughly the size of two products of single digits added together, so something in the 30s-60s is expected; 59 fits. The rule (top product + bottom product) reproduces 26, 33, and 40 exactly, so 59 for Group 4 is trustworthy.
Takeaway
When adding the numbers is too small, try multiplying them in pairs first — that pattern unlocks the center every time!
- Test a rule on Group 1
- Confirm the rule on Groups 2 and 3
- Apply the rule to Group 4