Patterns worksheet featuring geometric shapes with numbers, challenging students to find the missing values based on logical rules.

Let’s solve each pattern one by one. We’ll look at how the numbers are related in each shape and find the missing number.

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a) Cross shapes

First cross:
Top left = 7, Top right = 18, Bottom left = 9, Bottom right = 14
What’s in the middle? Nothing shown — but we need to find a rule that connects these four numbers to something. Wait — actually, looking again, there’s no center number given for the first two crosses. But in the third cross, we have a “?” on top left, and other numbers: top right=12, bottom left=6, bottom right=30.

Wait — maybe the pattern is between the pairs? Let’s check:

In first cross:
7 + 18 = 25
9 + 14 = 23 → Not equal.

7 × 2 = 14 (bottom right)
18 - 9 = 9 → not helpful.

Try this: Maybe top left + bottom right = top right + bottom left?

First cross: 7 + 14 = 21; 18 + 9 = 27 → No.

Another idea: Maybe multiply top left and bottom right, compare to top right and bottom left?

7 × 14 = 98
18 × 9 = 162 → No.

Wait — let’s look at second cross:

Top left=8, top right=10, bottom left=5, bottom right=16

8 + 16 = 24
10 + 5 = 15 → No.

But notice: In first cross: 7, 18, 9, 14
If we do: (top left × bottom right) = 7×14=98
(top right × bottom left)=18×9=162 → still no.

Wait — what if we add all four numbers?

First cross: 7+18+9+14 = 48
Second cross: 8+10+5+16 = 39
Third cross: ? +12+6+30 = ? + 48

No obvious pattern.

Alternative approach: Look at differences.

From first to second cross:

Top left: 7→8 (+1)
Top right: 18→10 (-8)
Bottom left: 9→5 (-4)
Bottom right: 14→16 (+2)

Not consistent.

Wait — perhaps the product of opposite corners?

First cross: 7×14 = 98, 18×9=162 → difference 64? Not useful.

Hold on — maybe it’s about the sum of top two vs bottom two?

First: top=7+18=25, bottom=9+14=23 → diff 2
Second: top=8+10=18, bottom=5+16=21 → diff -3 → no.

I think I’m overcomplicating.

Let me try this: In the first cross, 7 and 14 — 14 is 7×2
18 and 9 — 18 is 9×2

Oh! That’s interesting!

First cross:
Top left = 7, bottom right = 14 → 14 = 7 × 2
Top right = 18, bottom left = 9 → 18 = 9 × 2

Yes! So top right = bottom left × 2
And bottom right = top left × 2

Check second cross:
Top left = 8 → bottom right should be 8×2=16 → yes!
Top right = 10 → bottom left should be 10÷2=5 → yes!

Perfect!

So rule:
Bottom right = top left × 2
Top right = bottom left × 2

Now third cross:
We have top right = 12 → so bottom left = 12 ÷ 2 = 6 → matches given (6)
Bottom right = 30 → so top left = 30 ÷ 2 = 15

So the missing number (top left) is 15

✔ Answer for a): 15

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b) Teardrop shapes

Each has a big number on top, and two small circles below with numbers.

First: top=56, bottoms=7 and 8 → 7×8=56 → yes!

Second: top=36, bottoms=4 and 9 → 4×9=36 → yes!

Third: bottoms=12 and 12 → so top = 12×12 = 144

✔ Answer for b): 144

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c) Triangle with circles at corners and center

First triangle: corners 4, 2, 8 → center=48
4×2×8 = 64 → not 48

4+2+8=14 → no

Maybe (4×2) + (8×something)? No.

Wait: 4×2=8, then 8×6=48? Where does 6 come from?

Alternatively: 4×8=32, 2×8=16 → 32+16=48? Oh!

4×8 = 32
2×8 = 16
32 + 16 = 48 → but why multiply both by 8? 8 is one of the corners.

Actually: (left corner × bottom corner) + (right corner × bottom corner) = center?

Left=4, right=2, bottom=8 → (4×8)+(2×8)=32+16=48 → yes!

Factor out: 8×(4+2)=8×6=48 → same thing.

Second triangle: left=3, right=6, bottom=3 → center=27

(3×3) + (6×3) = 9 + 18 = 27 → yes!

Or 3×(3+6)=3×9=27 → yes!

So rule: bottom corner × (left + right) = center

Third triangle: left=7, right=2, bottom=2 → center=?

So: 2 × (7 + 2) = 2 × 9 = 18

✔ Answer for c): 18

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d) Three circles: one on top, two below

First: top=63, bottoms=4 and 9 → 4×9=36, not 63
4+9=13 → no

63 ÷ 9 = 7, 63 ÷ 4 = 15.75 → no

Wait: 4×9=36, 63-36=27 → not helpful.

Another idea: 4+9=13, 63÷13≈4.8 → no.

Look at second: top=53, bottoms=7 and 5 → 7×5=35, 53-35=18 → no

7+5=12, 53÷12≈4.4 → no

Wait — maybe it’s (left × right) + something?

First: 4×9=36, 63-36=27 → 27 is 3³? Not helpful.

Notice: 63 = 7×9, and 7 is... wait, 4 and 9 — 4+3=7? Arbitrary.

Another thought: Perhaps top = (left + right) × something?

First: 4+9=13, 63÷13 not integer.

Wait — let’s try subtraction: 9-4=5, 63÷5=12.6 → no.

Hold on — look at the numbers:

First: 4, 9 → 63
Second: 7, 5 → 53

What if: (left × 10) + right? 4×10+9=49 ≠63

(left + right) × 7? (4+9)×7=91≠63

Wait — 63 = 7×9, and 7 is not directly there.

But 4 and 9 — average is 6.5 → no.

Another idea: Maybe it’s left² + right²? 16+81=97≠63

Or |left - right| × something? |4-9|=5, 63÷5=12.6

This is tricky.

Let me try: For first, 4 and 9 → 4×9=36, then 63-36=27, which is 3×9 or 3×3×3.

For second: 7×5=35, 53-35=18, which is 2×9 or 3×6.

No clear link.

Wait — what if it’s (left + right) × k + m?

Set up equations:

For first: (4 + 9) * a + b = 63 → 13a + b = 63
Second: (7 + 5) * a + b = 53 → 12a + b = 53

Subtract: (13a+b) - (12a+b) = 63-53 → a=10

Then 12*10 + b = 53 → 120 + b = 53 → b= -67

Then for third: left=8, right=9 → sum=17 → 17*10 -67 = 170-67=103

But let’s verify with first: 13*10 -67=130-67=63 → yes
Second: 12*10-67=120-67=53 → yes

So rule: top = (left + right) × 10 - 67

That seems arbitrary, but mathematically works.

So for third: left=8, right=9 → sum=17 → 17×10 -67 = 170-67=103

But is there a better way? Maybe I missed something simpler.

Alternative: Notice 63 and 53 — difference of 10, while sums are 13 and 12 — difference of 1, so slope 10, as above.

Perhaps it’s designed that way.

Maybe it’s (left × right) + (left + right) × something.

4×9=36, 4+9=13, 36 + 27=63, 27=13×2 +1? Messy.

I think the linear relation is correct: top = 10*(left + right) - 67

So for 8 and 9: 10*(17) -67 = 170-67=103

✔ Answer for d): 103

But let me double-check if there's a multiplicative pattern.

Another idea: 63 = 7*9, and 7 = 4 + 3? Not consistent.

53 is prime, so hard to factor.

I think 103 is correct based on the pattern.

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e) Rectangles with three numbers

First: 16, 8, 4 → each divided by 2: 16÷2=8, 8÷2=4

Second: 36, 18, 9 → 36÷2=18, 18÷2=9

So pattern: each number is half the previous.

Third: starts with 16, so next should be 16÷2=8, then 8÷2=4

So missing numbers: 8 and 4

✔ Answer for e): 8 and 4

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Now, compiling all answers:

a) 15
b) 144
c) 18
d) 103
e) 8 and 4

Final Answer:
a) 15
b) 144
c) 18
d) 103
e) 8, 4