Free and fun math worksheets with puzzles and riddles - Free Printable
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Step-by-step solution for: Free and fun math worksheets with puzzles and riddles
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Step-by-step solution for: Free and fun math worksheets with puzzles and riddles
To find the missing number in the center of the first shape, we need to discover the pattern used in the second shape.
Step 1: Analyze the second shape (the star)
Let's look at the numbers on the outside and how they relate to the center number.
The outer numbers are: 8, 10, 20, 60, 40, 2, 4, 12.
The center number is empty in the image provided for the second shape? Wait, looking closely at the second image, the center circle is empty. Let me re-read the prompt. Ah, usually in these problems, one part has a known center to establish the rule, or the rule applies to both. Let's look at the first image again. The first image has an empty center. The second image also has an empty center. This is unusual.
Let me re-examine the images carefully.
Image 1: A square with a diamond inside. Outer corners have numbers: 6, 2, 3, 3, 1, 2, 2, 3? No, let's trace the connections.
There are 8 small circles on the perimeter of the inner diamond/square structure.
Top-left: 6
Top-right: 2
Right-top: 2
Right-bottom: 3
Bottom-right: 3
Bottom-left: 1
Left-bottom: 2
Left-top: 2
Wait, let's look at the lines.
In the first figure, there are 8 numbers surrounding the center: 6, 2, 2, 3, 3, 1, 2, 2.
Actually, let's look at the pairs opposite each other or adjacent.
Let's look at the second figure (star). It has 8 numbers: 8, 10, 20, 60, 40, 2, 4, 12.
Is there a relationship between the two figures? Maybe the numbers correspond?
No, usually each figure is independent, or they share a common rule. Since both centers are blank, that can't be right. One of them must have a solution derived from the other, or I am misinterpreting the "blank" center.
Let's look really closely at the second image. The center circle is just white. The first image's center circle is just white.
Is it possible the question asks to find the number for the *first* diagram based on a rule found in the *second*? But the second diagram doesn't have a center number either.
Let me reconsider the standard types of these puzzles.
Often, the sum of opposite numbers is constant.
Or the product.
Or sum of all outer numbers equals the center.
Let's test "Sum of all outer numbers" on Figure 2:
$8 + 10 + 20 + 60 + 40 + 2 + 4 + 12 = ?$
$8+12=20$
$10+2=12$ -> $20+12=32$
$20+40=60$ -> $32+60=92$
$60+4=64$ -> $92+64=156$
Total = 156.
Let's test "Sum of all outer numbers" on Figure 1:
Numbers: 6, 2, 2, 3, 3, 1, 2, 2.
$6+2+2+3+3+1+2+2 = ?$
$6+2=8$
$8+2=10$
$10+3=13$
$13+3=16$
$16+1=17$
$17+2=19$
$19+2=21$
Total = 21.
This doesn't seem to link them directly unless the center numbers are related. But both are missing.
Let's look for another pattern. Maybe opposite pairs?
Figure 2 Opposites:
8 and 40? Sum = 48. Product = 320.
10 and 2? Sum = 12. Product = 20.
20 and 4? Sum = 24. Product = 80.
60 and 12? Sum = 72. Product = 720.
No obvious constant sum or product.
Let's look at adjacent pairs or groups.
Maybe the rule is: Sum of opposite pairs is equal?
Fig 2:
$8+40=48$
$10+2=12$
$20+4=24$
$60+12=72$
No.
Maybe Product of opposite pairs?
$8 \times 40 = 320$
$10 \times 2 = 20$
$20 \times 4 = 80$
$60 \times 12 = 720$
No.
Let's look at the structure again.
Figure 1: The numbers are on vertices of a square and midpoints of sides?
Top-Left corner: 6
Top side midpoint: 2
Top-Right corner: 2
Right side midpoint: 3
Bottom-Right corner: 3
Bottom side midpoint: 1
Bottom-Left corner: 2
Left side midpoint: 2
Figure 2: Star with 8 points.
Top: 8
Top-Right: 10
Right: 20
Bottom-Right: 60
Bottom: 40
Bottom-Left: 2
Left: 4
Top-Left: 12
Is there a connection between Fig 1 and Fig 2 positions?
If we map Fig 1 to Fig 2 clockwise starting from top/top-left:
Fig 1 Top-Left (6) -> Fig 2 Top-Left (12)? Ratio 2.
Fig 1 Top-Mid (2) -> Fig 2 Top (8)? Ratio 4.
Fig 1 Top-Right (2) -> Fig 2 Top-Right (10)? Ratio 5.
Fig 1 Right-Mid (3) -> Fig 2 Right (20)? Ratio ~6.6.
This seems unlikely.
Let's rethink the problem source. Often in these tests, if two diagrams are presented side-by-side, they might follow the same internal logic, and one of them might have a typo in my perception, OR the question implies finding the center for *both* or the center is the same value? No, the numbers are different.
Wait, look at the second diagram again. Is it possible the center number is not missing, but the question is asking for the center of the first diagram, implying the rule is derived from the second diagram where the center is somehow calculable or given?
Actually, looking at similar online puzzles, sometimes the "center" is the result of an operation on the outer numbers. If both centers are blank, maybe the task is to find the number that *would* go in the center of the first diagram, assuming the same rule applies to both, but we need a reference.
Is it possible I missed a number?
Let me check the second image very carefully.
Center is empty.
First image center is empty.
Could the rule be related to opposite numbers adding up to the same total?
Let's check Figure 1 opposites.
Top-Left (6) vs Bottom-Right (3). Sum = 9.
Top-Mid (2) vs Bottom-Mid (1). Sum = 3.
Top-Right (2) vs Bottom-Left (2). Sum = 4.
Right-Mid (3) vs Left-Mid (2). Sum = 5.
Sums: 9, 3, 4, 5. No pattern.
Let's check products for Figure 1 opposites.
$6 \times 3 = 18$
$2 \times 1 = 2$
$2 \times 2 = 4$
$3 \times 2 = 6$
No.
Let's try sum of all numbers divided by something.
Fig 1 Sum = 21.
Fig 2 Sum = 156.
Let's try sum of alternating numbers.
Fig 1:
Corners: $6 + 2 + 3 + 2 = 13$.
Mids: $2 + 3 + 1 + 2 = 8$.
Difference: $13 - 8 = 5$.
Sum: $13 + 8 = 21$.
Fig 2:
Let's split into two sets of 4 (e.g., cardinal and intercardinal).
Cardinal (Top, Right, Bottom, Left): $8 + 20 + 40 + 4 = 72$.
Intercardinal (TR, BR, BL, TL): $10 + 60 + 2 + 12 = 84$.
Difference: $84 - 72 = 12$.
Sum: $156$.
Is there a relationship between the sums/differences?
Fig 1: Corner Sum 13, Mid Sum 8.
Fig 2: Group A Sum 72, Group B Sum 84.
Let's look at pairs of adjacent numbers.
Fig 2:
$8+10=18$
$10+20=30$
$20+60=80$
$60+40=100$
$40+2=42$
$2+4=6$
$4+12=16$
$12+8=20$
Let's look at products of opposite numbers again for Fig 2.
$8 \times 40 = 320$
$20 \times 4 = 80$
$10 \times 2 = 20$
$60 \times 12 = 720$
Notice: $320 / 4 = 80$. $720 / 36 = 20$? No.
Notice: $8 \times 40 = 320$. $20 \times 4 = 80$. $320 / 80 = 4$.
$60 \times 12 = 720$. $10 \times 2 = 20$. $720 / 20 = 36$.
Let's try a different perspective. What if the center number is the sum of the outer numbers divided by the count?
Fig 1: $21 / 8 = 2.625$. Unlikely.
Fig 2: $156 / 8 = 19.5$. Unlikely.
What if the center is the difference between the largest and smallest?
Fig 1: Max 6, Min 1. Diff = 5.
Fig 2: Max 60, Min 2. Diff = 58.
What if the center is the product of specific numbers?
Let's look at the visual structure again.
Figure 1: Square inside a square. The numbers are at the vertices of the inner diamond and the midpoints of the inner diamond's sides? No, the lines connect the outer square corners to the inner diamond vertices?
Actually, the lines form a grid.
Let's label positions 1-8 clockwise from top-left for Fig 1.
1: 6 (Corner)
2: 2 (Side)
3: 2 (Corner) -- Wait, looking at the image, the circle with '2' is on the top edge of the diamond? Or is it a corner of the outer square?
The diagram shows a large square. Inside is a diamond (rotated square).
The circles are located at:
- The 4 corners of the inner diamond.
- The 4 midpoints of the sides of the inner diamond.
Let's verify the values based on this geometry.
Top Vertex of Diamond: Circle with 2? No, the circle is outside the diamond line?
Let's trace the lines.
There is an outer square boundary.
There is an inner diamond boundary.
The circles are placed on the vertices of the inner diamond and the midpoints of the inner diamond's sides?
Actually, looking at the left image:
The circle with '6' is at the top-left corner of the *outer* square? No, it's near the corner.
The circle with '2' (top) is on the top side of the inner diamond?
Let's assume the standard 8-position clock face layout.
Pos 12 o'clock: 2
Pos 1:30: 2
Pos 3:00: 3
Pos 4:30: 3
Pos 6:00: 1
Pos 7:30: 2
Pos 9:00: 2
Pos 10:30: 6
Let's assume the same for the right image (Star).
Pos 12: 8
Pos 1:30: 10
Pos 3: 20
Pos 4:30: 60
Pos 6: 40
Pos 7:30: 2
Pos 9: 4
Pos 10:30: 12
Okay, now we have two sets of 8 numbers.
Set 1: $\{2, 2, 3, 3, 1, 2, 2, 6\}$
Set 2: $\{8, 10, 20, 60, 40, 2, 4, 12\}$
Is there a mathematical operation linking Set 1 to Set 2?
$2 \rightarrow 8$ ($\times 4$)
$2 \rightarrow 10$ ($\times 5$)
$3 \rightarrow 20$ ($\times 6.6$)
$3 \rightarrow 60$ ($\times 20$)
$1 \rightarrow 40$ ($\times 40$)
$2 \rightarrow 2$ ($\times 1$)
$2 \rightarrow 4$ ($\times 2$)
$6 \rightarrow 12$ ($\times 2$)
No consistent multiplier.
However, notice that $6 \times 2 = 12$. (Pos 10:30)
$2 \times 2 = 4$. (Pos 9:00)
$2 \times 1 = 2$. (Pos 7:30)
$1 \times 40 = 40$. (Pos 6:00) -- Breaks pattern.
$3 \times 20 = 60$. (Pos 4:30)
$3 \times 6.6 = 20$.
$2 \times 5 = 10$.
$2 \times 4 = 8$.
Let's look at the products of opposite numbers in each figure.
Figure 1:
Opposite pairs (across the center):
- Top (2) and Bottom (1): $2 \times 1 = 2$
- Top-Right (2) and Bottom-Left (2): $2 \times 2 = 4$
- Right (3) and Left (2): $3 \times 2 = 6$
- Bottom-Right (3) and Top-Left (6): $3 \times 6 = 18$
Products: 2, 4, 6, 18.
Figure 2:
Opposite pairs:
- Top (8) and Bottom (40): $8 \times 40 = 320$
- Top-Right (10) and Bottom-Left (2): $10 \times 2 = 20$
- Right (20) and Left (4): $20 \times 4 = 80$
- Bottom-Right (60) and Top-Left (12): $60 \times 12 = 720$
Products: 320, 20, 80, 720.
Is there a relationship between the products of Fig 1 and Fig 2?
Fig 1 Products: 2, 4, 6, 18
Fig 2 Products: 320, 20, 80, 720
Let's check ratios:
$320 / 2 = 160$
$20 / 4 = 5$
$80 / 6 = 13.33$
$720 / 18 = 40$
No.
Let's check sums of opposite pairs again.
Fig 1 Sums:
$2+1=3$
$2+2=4$
$3+2=5$
$3+6=9$
Sums: 3, 4, 5, 9.
Fig 2 Sums:
$8+40=48$
$10+2=12$
$20+4=24$
$60+12=72$
Sums: 48, 12, 24, 72.
Look at the Fig 2 sums: 12, 24, 48, 72.
$12 \times 1 = 12$
$12 \times 2 = 24$
$12 \times 4 = 48$
$12 \times 6 = 72$
Multipliers: 1, 2, 4, 6.
Now look at Fig 1 sums: 3, 4, 5, 9.
Do these match the multipliers 1, 2, 4, 6? No.
But wait.
Pair (Top-Right/Bottom-Left) in Fig 2 had Sum 12. Multiplier 1.
Pair (Right/Left) in Fig 2 had Sum 24. Multiplier 2.
Pair (Top/Bottom) in Fig 2 had Sum 48. Multiplier 4.
Pair (Bottom-Right/Top-Left) in Fig 2 had Sum 72. Multiplier 6.
Let's see if the Fig 1 sums correspond to these positions.
Position: Top-Right/Bottom-Left. Fig 1 Sum = $2+2=4$.
Position: Right/Left. Fig 1 Sum = $3+2=5$.
Position: Top/Bottom. Fig 1 Sum = $2+1=3$.
Position: Bottom-Right/Top-Left. Fig 1 Sum = $3+6=9$.
Sums: 4, 5, 3, 9.
Multipliers needed for Fig 2 were: 1, 2, 4, 6.
No direct match.
However, look at the products of the Fig 1 sums?
$3 \times 4 \times 5 \times 9 = 540$.
$12 \times 24 \times 48 \times 72 = \dots$ huge.
Let's step back. What is the most common simple logic for these "find the center number" puzzles?
1. Sum of all outer numbers.
2. Sum of opposite pairs is constant.
3. (Sum of some) - (Sum of others) = Center.
4. Product of some / Product of others.
Let's try (Sum of Even Positions) - (Sum of Odd Positions).
Fig 2:
Even pos (12, 3, 6, 9 o'clock approx?): 8, 20, 40, 4. Sum = 72.
Odd pos (1:30, 4:30, 7:30, 10:30): 10, 60, 2, 12. Sum = 84.
Diff = 12.
Fig 1:
Even pos (Top, Right, Bottom, Left): 2, 3, 1, 2. Sum = 8.
Odd pos (TR, BR, BL, TL): 2, 3, 2, 6. Sum = 13.
Diff = $13 - 8 = 5$.
So, if the rule is "Difference between the sum of diagonal/intercardinal numbers and cardinal numbers", then:
Fig 2 Center would be 12 (or -12).
Fig 1 Center would be 5 (or -5).
Is it possible the second image *has* a center number that I'm supposed to infer, or is the "12" in the difference significant?
Wait, look at the numbers in Fig 2 again.
12, 24, 48, 72 are the sums of opposite pairs.
$12 = 12 \times 1$
$24 = 12 \times 2$
$48 = 12 \times 4$
$72 = 12 \times 6$
The multipliers are 1, 2, 4, 6.
Sum of multipliers = $1+2+4+6 = 13$.
Product? $1 \times 2 \times 4 \times 6 = 48$.
Now look at Fig 1 opposite sums: 3, 4, 5, 9.
Is there a base number here?
GCD of 3,4,5,9 is 1.
Not helpful.
Let's try another very common pattern: Center = Sum of Outer Numbers / 2 or similar.
Fig 2 Sum = 156. $156 / ? = $ Integer?
$156 / 12 = 13$.
$156 / 13 = 12$.
Fig 1 Sum = 21.
$21 / ? = $ Integer?
$21 / 3 = 7$.
$21 / 7 = 3$.
Let's look at the relationship between the two figures again.
Maybe the center of Fig 1 is related to the center of Fig 2?
If the rule is consistent, and we assume the "Difference of alternating sums" rule:
Fig 2 Diff = 12.
Fig 1 Diff = 5.
Is "5" a plausible answer?
Let's check if there's a simpler additive rule.
Fig 2: $8+10+20+60+40+2+4+12 = 156$.
Fig 1: $6+2+2+3+3+1+2+2 = 21$.
What if the center is the sum of the digits?
Fig 2: $1+5+6 = 12$.
Fig 1: $2+1 = 3$.
What if the center is the number of letters in the English word for the sum? No.
Let's look at the visual arrangement of Fig 1 again.
It looks like the numbers are grouped.
Top half: 6, 2, 2. Sum = 10.
Bottom half: 2, 1, 3, 3, 2? No.
Left half: 6, 2, 2, 1. Sum = 11.
Right half: 2, 3, 3, 2. Sum = 10.
Diff = 1.
Top half (above horizontal midline): 6, 2, 2. (Assuming 6 is TL, 2 is T, 2 is TR). Sum = 10.
Bottom half: 2, 1, 3, 3? (BL, B, BR, R??).
Let's define halves strictly.
Horizontal split:
Top: 6 (TL), 2 (T), 2 (TR). Sum = 10.
Bottom: 2 (L), 1 (B), 3 (BR), 3 (R)? No, L and R are on the midline.
Usually "Top" includes Top-Left, Top, Top-Right.
"Bottom" includes Bottom-Left, Bottom, Bottom-Right.
What about Left and Right?
If we do Vertical Split:
Left: 6 (TL), 2 (L), 2 (BL). Sum = 10.
Right: 2 (TR), 3 (R), 3 (BR). Sum = 8.
Top: 6 (TL), 2 (T), 2 (TR). Sum = 10.
Bottom: 2 (BL), 1 (B), 3 (BR). Sum = 6.
Left+Right = 18.
Top+Bottom = 16.
Remaining numbers: L(2) and R(3).
This is getting messy.
Let's go back to the Opposite Pairs Sum idea for Fig 2.
Sums: 12, 24, 48, 72.
These are all multiples of 12.
$1 \times 12, 2 \times 12, 4 \times 12, 6 \times 12$.
The coefficients are 1, 2, 4, 6.
Note that $1+2+4+6 = 13$.
Also note that the numbers in the "1" coefficient pair are 10 and 2.
"2" coeff pair: 20 and 4.
"4" coeff pair: 8 and 40.
"6" coeff pair: 60 and 12.
Is there a pattern in the coefficients relative to position?
Pos 1:30 (10,2) -> Coeff 1.
Pos 3:00 (20,4) -> Coeff 2.
Pos 12:00 (8,40) -> Coeff 4.
Pos 4:30 (60,12) -> Coeff 6.
Now apply this "Coefficient" logic to Fig 1?
We need to find a "Base Number" for Fig 1 such that the opposite sums are multiples of it.
Fig 1 Opposite Sums: 3, 4, 5, 9.
GCD(3,4,5,9) = 1.
So Base = 1.
Coefficients: 3, 4, 5, 9.
Sum of Coefficients = $3+4+5+9 = 21$.
Does the center equal the sum of coefficients?
If so, Fig 2 Center = 13.
Fig 1 Center = 21.
Let's check if 13 makes sense for Fig 2.
Sum of outer numbers = 156.
$156 / 12 = 13$.
So, Center = Sum of Outer Numbers / Base Multiple.
For Fig 2: Base Multiple was 12. Sum = 156. Center = 13.
For Fig 1: Base Multiple was 1. Sum = 21. Center = 21.
Alternatively, Center = Sum of Coefficients.
Fig 2 Coeffs: 1,2,4,6. Sum = 13.
Fig 1 Coeffs: 3,4,5,9. Sum = 21.
This seems like a strong candidate. The logic is:
1. Calculate sum of opposite pairs.
2. Find the Greatest Common Divisor (GCD) of these sums. This is the "Base".
3. Divide each pair sum by the Base to get "Coefficients".
4. The Center Number is the Sum of these Coefficients.
Let's double check this logic.
Why would the center be the sum of coefficients?
In Fig 2, Sum of Outer = 156.
Sum of Pair Sums = $12+24+48+72 = 156$.
Sum of Pair Sums = Base $\times$ (Sum of Coeffs).
$156 = 12 \times (\text{Sum of Coeffs})$.
$\text{Sum of Coeffs} = 156 / 12 = 13$.
So, Center = Sum of Outer Numbers / GCD of Opposite Pair Sums.
Let's apply this formula to Fig 1.
1. Opposite Pair Sums: 3, 4, 5, 9.
2. GCD(3, 4, 5, 9) = 1.
3. Sum of Outer Numbers = 21.
4. Center = $21 / 1 = 21$.
Let's try one more hypothesis. What if the center is simply the Sum of the Outer Numbers for Fig 1, because the GCD is 1?
And for Fig 2, the center is 13?
Is there any other interpretation?
What if the rule is simpler?
Look at Fig 2:
$8+10+20+60+40+2+4+12 = 156$.
$1+5+6=12$? No.
$156 / 12 = 13$.
Look at Fig 1:
$6+2+2+3+3+1+2+2 = 21$.
$21 / 1 = 21$.
Is it possible the answer is just the sum?
If the GCD was always 1, yes. But in Fig 2, the GCD is 12.
Why divide by GCD?
Maybe the rule is: Center = Sum of Opposite Pair Sums / Largest Pair Sum?
Fig 2: $156 / 72 = 2.16$. No.
Center = Sum of Opposite Pair Sums / Smallest Pair Sum?
Fig 2: $156 / 12 = 13$.
Fig 1: $21 / 3 = 7$.
Let's compare "Divide by GCD" vs "Divide by Smallest Sum".
In Fig 2, GCD=12, Smallest Sum=12. They are the same.
In Fig 1, GCD=1, Smallest Sum=3.
If rule is "Divide by Smallest Sum": Answer = 7.
If rule is "Divide by GCD": Answer = 21.
Which is more "standard"?
Usually, these puzzles rely on integer properties.
GCD is a more robust mathematical property than "Smallest Sum".
However, 21 is the total sum. If the answer is just the total sum, why the complex structure in Fig 2?
In Fig 2, if the answer was the total sum, it would be 156.
If the answer is 13, it's a derived number.
Let's look for clues in the numbers themselves.
Fig 1: 1, 2, 2, 2, 2, 3, 3, 6.
Fig 2: 2, 4, 8, 10, 12, 20, 40, 60.
Notice Fig 2 numbers are all even.
Fig 1 has odd numbers (1, 3, 3).
Let's try Average of Opposite Sums.
Fig 2: $(12+24+48+72)/4 = 156/4 = 39$.
Fig 1: $(3+4+5+9)/4 = 21/4 = 5.25$. No.
Let's try Sum of Unique Digits?
No.
Let's reconsider the Difference of Alternating Sums.
Fig 2: $|84 - 72| = 12$.
Fig 1: $|13 - 8| = 5$.
Between 21, 7, and 5, which is the best fit?
The "GCD" method yielded 21.
The "Smallest Sum" method yielded 7.
The "Alternating Difference" method yielded 5.
Let's look at the shape again.
Fig 1 is a square/diamond.
Fig 2 is a star.
Often, the number of points matters. Both have 8.
Let's try one more calculation for Fig 2.
$(8 \times 40) / (10 \times 2) = 320 / 20 = 16$.
$(60 \times 12) / (20 \times 4) = 720 / 80 = 9$.
$16 - 9 = 7$?
$16 + 9 = 25$?
Let's try for Fig 1.
$(2 \times 1) / (2 \times 2) = 2 / 4 = 0.5$.
$(3 \times 6) / (3 \times 2) = 18 / 6 = 3$.
$3 - 0.5 = 2.5$.
$3 + 0.5 = 3.5$.
No.
Let's go with the most mathematically elegant solution found:
Rule: The center number is the sum of the outer numbers divided by the greatest common divisor (GCD) of the sums of opposite pairs.
Verification for Figure 2:
- Opposite sums: 12, 24, 48, 72.
- GCD(12, 24, 48, 72) = 12.
- Total Sum = 156.
- Center = $156 / 12 = 13$.
Application to Figure 1:
- Opposite sums: 3, 4, 5, 9.
- GCD(3, 4, 5, 9) = 1.
- Total Sum = 21.
- Center = $21 / 1 = 21$.
Alternative Check:
Is there a simpler rule?
Sum of outer numbers for Fig 1 is 21.
Sum of outer numbers for Fig 2 is 156.
If the answer for Fig 2 was 156, the rule would be "Sum".
But 156 is a large number. 13 is "cleaner".
If the answer for Fig 1 is 21, it is just the sum.
Let's consider if "5" (from alternating difference) is better.
Fig 2 Diff = 12.
Fig 1 Diff = 5.
Is 12 a "special" number in Fig 2? Yes, it's the GCD and the smallest pair sum.
Is 5 a "special" number in Fig 1?
Smallest pair sum is 3. GCD is 1.
The alternating difference is 5.
Comparing "13" vs "12" for Fig 2.
If the center of Fig 2 was 12, the rule would be "Alternating Difference".
If the center of Fig 2 was 13, the rule would be "Sum / GCD".
Without the center of Fig 2 provided, we have to guess the intended complexity.
"Alternating Difference" is a very common puzzle trope.
"Sum / GCD" is rare.
Let's look at the numbers in Fig 1 again.
6, 2, 2, 3, 3, 1, 2, 2.
Alternating Sum 1: $6+2+3+2 = 13$.
Alternating Sum 2: $2+3+1+2 = 8$.
Diff = 5.
Let's look at Fig 2 again.
Alternating Sum 1: $12+10+60+2 = 84$. (Starting TL, going clockwise: TL, TR, BR, BL? No, TL, TR, BR, BL are not alternating indices. Indices 1,3,5,7 vs 2,4,6,8).
Let's use indices 1-8 clockwise from Top.
1:8, 2:10, 3:20, 4:60, 5:40, 6:2, 7:4, 8:12.
Odd indices (1,3,5,7): $8+20+40+4 = 72$.
Even indices (2,4,6,8): $10+60+2+12 = 84$.
Diff = 12.
If the rule is Absolute Difference of Alternating Sums, then:
Fig 2 Center = 12.
Fig 1 Center = 5.
This feels like a very standard IQ test pattern. The "Sum/GCD" is too complex. The "Alternating Difference" is simple and uses basic arithmetic.
Final Answer Prediction: 5.
──────────────────────────────────────
Final Answer:
5
Step 1: Analyze the second shape (the star)
Let's look at the numbers on the outside and how they relate to the center number.
The outer numbers are: 8, 10, 20, 60, 40, 2, 4, 12.
The center number is empty in the image provided for the second shape? Wait, looking closely at the second image, the center circle is empty. Let me re-read the prompt. Ah, usually in these problems, one part has a known center to establish the rule, or the rule applies to both. Let's look at the first image again. The first image has an empty center. The second image also has an empty center. This is unusual.
Let me re-examine the images carefully.
Image 1: A square with a diamond inside. Outer corners have numbers: 6, 2, 3, 3, 1, 2, 2, 3? No, let's trace the connections.
There are 8 small circles on the perimeter of the inner diamond/square structure.
Top-left: 6
Top-right: 2
Right-top: 2
Right-bottom: 3
Bottom-right: 3
Bottom-left: 1
Left-bottom: 2
Left-top: 2
Wait, let's look at the lines.
In the first figure, there are 8 numbers surrounding the center: 6, 2, 2, 3, 3, 1, 2, 2.
Actually, let's look at the pairs opposite each other or adjacent.
Let's look at the second figure (star). It has 8 numbers: 8, 10, 20, 60, 40, 2, 4, 12.
Is there a relationship between the two figures? Maybe the numbers correspond?
No, usually each figure is independent, or they share a common rule. Since both centers are blank, that can't be right. One of them must have a solution derived from the other, or I am misinterpreting the "blank" center.
Let's look really closely at the second image. The center circle is just white. The first image's center circle is just white.
Is it possible the question asks to find the number for the *first* diagram based on a rule found in the *second*? But the second diagram doesn't have a center number either.
Let me reconsider the standard types of these puzzles.
Often, the sum of opposite numbers is constant.
Or the product.
Or sum of all outer numbers equals the center.
Let's test "Sum of all outer numbers" on Figure 2:
$8 + 10 + 20 + 60 + 40 + 2 + 4 + 12 = ?$
$8+12=20$
$10+2=12$ -> $20+12=32$
$20+40=60$ -> $32+60=92$
$60+4=64$ -> $92+64=156$
Total = 156.
Let's test "Sum of all outer numbers" on Figure 1:
Numbers: 6, 2, 2, 3, 3, 1, 2, 2.
$6+2+2+3+3+1+2+2 = ?$
$6+2=8$
$8+2=10$
$10+3=13$
$13+3=16$
$16+1=17$
$17+2=19$
$19+2=21$
Total = 21.
This doesn't seem to link them directly unless the center numbers are related. But both are missing.
Let's look for another pattern. Maybe opposite pairs?
Figure 2 Opposites:
8 and 40? Sum = 48. Product = 320.
10 and 2? Sum = 12. Product = 20.
20 and 4? Sum = 24. Product = 80.
60 and 12? Sum = 72. Product = 720.
No obvious constant sum or product.
Let's look at adjacent pairs or groups.
Maybe the rule is: Sum of opposite pairs is equal?
Fig 2:
$8+40=48$
$10+2=12$
$20+4=24$
$60+12=72$
No.
Maybe Product of opposite pairs?
$8 \times 40 = 320$
$10 \times 2 = 20$
$20 \times 4 = 80$
$60 \times 12 = 720$
No.
Let's look at the structure again.
Figure 1: The numbers are on vertices of a square and midpoints of sides?
Top-Left corner: 6
Top side midpoint: 2
Top-Right corner: 2
Right side midpoint: 3
Bottom-Right corner: 3
Bottom side midpoint: 1
Bottom-Left corner: 2
Left side midpoint: 2
Figure 2: Star with 8 points.
Top: 8
Top-Right: 10
Right: 20
Bottom-Right: 60
Bottom: 40
Bottom-Left: 2
Left: 4
Top-Left: 12
Is there a connection between Fig 1 and Fig 2 positions?
If we map Fig 1 to Fig 2 clockwise starting from top/top-left:
Fig 1 Top-Left (6) -> Fig 2 Top-Left (12)? Ratio 2.
Fig 1 Top-Mid (2) -> Fig 2 Top (8)? Ratio 4.
Fig 1 Top-Right (2) -> Fig 2 Top-Right (10)? Ratio 5.
Fig 1 Right-Mid (3) -> Fig 2 Right (20)? Ratio ~6.6.
This seems unlikely.
Let's rethink the problem source. Often in these tests, if two diagrams are presented side-by-side, they might follow the same internal logic, and one of them might have a typo in my perception, OR the question implies finding the center for *both* or the center is the same value? No, the numbers are different.
Wait, look at the second diagram again. Is it possible the center number is not missing, but the question is asking for the center of the first diagram, implying the rule is derived from the second diagram where the center is somehow calculable or given?
Actually, looking at similar online puzzles, sometimes the "center" is the result of an operation on the outer numbers. If both centers are blank, maybe the task is to find the number that *would* go in the center of the first diagram, assuming the same rule applies to both, but we need a reference.
Is it possible I missed a number?
Let me check the second image very carefully.
Center is empty.
First image center is empty.
Could the rule be related to opposite numbers adding up to the same total?
Let's check Figure 1 opposites.
Top-Left (6) vs Bottom-Right (3). Sum = 9.
Top-Mid (2) vs Bottom-Mid (1). Sum = 3.
Top-Right (2) vs Bottom-Left (2). Sum = 4.
Right-Mid (3) vs Left-Mid (2). Sum = 5.
Sums: 9, 3, 4, 5. No pattern.
Let's check products for Figure 1 opposites.
$6 \times 3 = 18$
$2 \times 1 = 2$
$2 \times 2 = 4$
$3 \times 2 = 6$
No.
Let's try sum of all numbers divided by something.
Fig 1 Sum = 21.
Fig 2 Sum = 156.
Let's try sum of alternating numbers.
Fig 1:
Corners: $6 + 2 + 3 + 2 = 13$.
Mids: $2 + 3 + 1 + 2 = 8$.
Difference: $13 - 8 = 5$.
Sum: $13 + 8 = 21$.
Fig 2:
Let's split into two sets of 4 (e.g., cardinal and intercardinal).
Cardinal (Top, Right, Bottom, Left): $8 + 20 + 40 + 4 = 72$.
Intercardinal (TR, BR, BL, TL): $10 + 60 + 2 + 12 = 84$.
Difference: $84 - 72 = 12$.
Sum: $156$.
Is there a relationship between the sums/differences?
Fig 1: Corner Sum 13, Mid Sum 8.
Fig 2: Group A Sum 72, Group B Sum 84.
Let's look at pairs of adjacent numbers.
Fig 2:
$8+10=18$
$10+20=30$
$20+60=80$
$60+40=100$
$40+2=42$
$2+4=6$
$4+12=16$
$12+8=20$
Let's look at products of opposite numbers again for Fig 2.
$8 \times 40 = 320$
$20 \times 4 = 80$
$10 \times 2 = 20$
$60 \times 12 = 720$
Notice: $320 / 4 = 80$. $720 / 36 = 20$? No.
Notice: $8 \times 40 = 320$. $20 \times 4 = 80$. $320 / 80 = 4$.
$60 \times 12 = 720$. $10 \times 2 = 20$. $720 / 20 = 36$.
Let's try a different perspective. What if the center number is the sum of the outer numbers divided by the count?
Fig 1: $21 / 8 = 2.625$. Unlikely.
Fig 2: $156 / 8 = 19.5$. Unlikely.
What if the center is the difference between the largest and smallest?
Fig 1: Max 6, Min 1. Diff = 5.
Fig 2: Max 60, Min 2. Diff = 58.
What if the center is the product of specific numbers?
Let's look at the visual structure again.
Figure 1: Square inside a square. The numbers are at the vertices of the inner diamond and the midpoints of the inner diamond's sides? No, the lines connect the outer square corners to the inner diamond vertices?
Actually, the lines form a grid.
Let's label positions 1-8 clockwise from top-left for Fig 1.
1: 6 (Corner)
2: 2 (Side)
3: 2 (Corner) -- Wait, looking at the image, the circle with '2' is on the top edge of the diamond? Or is it a corner of the outer square?
The diagram shows a large square. Inside is a diamond (rotated square).
The circles are located at:
- The 4 corners of the inner diamond.
- The 4 midpoints of the sides of the inner diamond.
Let's verify the values based on this geometry.
Top Vertex of Diamond: Circle with 2? No, the circle is outside the diamond line?
Let's trace the lines.
There is an outer square boundary.
There is an inner diamond boundary.
The circles are placed on the vertices of the inner diamond and the midpoints of the inner diamond's sides?
Actually, looking at the left image:
The circle with '6' is at the top-left corner of the *outer* square? No, it's near the corner.
The circle with '2' (top) is on the top side of the inner diamond?
Let's assume the standard 8-position clock face layout.
Pos 12 o'clock: 2
Pos 1:30: 2
Pos 3:00: 3
Pos 4:30: 3
Pos 6:00: 1
Pos 7:30: 2
Pos 9:00: 2
Pos 10:30: 6
Let's assume the same for the right image (Star).
Pos 12: 8
Pos 1:30: 10
Pos 3: 20
Pos 4:30: 60
Pos 6: 40
Pos 7:30: 2
Pos 9: 4
Pos 10:30: 12
Okay, now we have two sets of 8 numbers.
Set 1: $\{2, 2, 3, 3, 1, 2, 2, 6\}$
Set 2: $\{8, 10, 20, 60, 40, 2, 4, 12\}$
Is there a mathematical operation linking Set 1 to Set 2?
$2 \rightarrow 8$ ($\times 4$)
$2 \rightarrow 10$ ($\times 5$)
$3 \rightarrow 20$ ($\times 6.6$)
$3 \rightarrow 60$ ($\times 20$)
$1 \rightarrow 40$ ($\times 40$)
$2 \rightarrow 2$ ($\times 1$)
$2 \rightarrow 4$ ($\times 2$)
$6 \rightarrow 12$ ($\times 2$)
No consistent multiplier.
However, notice that $6 \times 2 = 12$. (Pos 10:30)
$2 \times 2 = 4$. (Pos 9:00)
$2 \times 1 = 2$. (Pos 7:30)
$1 \times 40 = 40$. (Pos 6:00) -- Breaks pattern.
$3 \times 20 = 60$. (Pos 4:30)
$3 \times 6.6 = 20$.
$2 \times 5 = 10$.
$2 \times 4 = 8$.
Let's look at the products of opposite numbers in each figure.
Figure 1:
Opposite pairs (across the center):
- Top (2) and Bottom (1): $2 \times 1 = 2$
- Top-Right (2) and Bottom-Left (2): $2 \times 2 = 4$
- Right (3) and Left (2): $3 \times 2 = 6$
- Bottom-Right (3) and Top-Left (6): $3 \times 6 = 18$
Products: 2, 4, 6, 18.
Figure 2:
Opposite pairs:
- Top (8) and Bottom (40): $8 \times 40 = 320$
- Top-Right (10) and Bottom-Left (2): $10 \times 2 = 20$
- Right (20) and Left (4): $20 \times 4 = 80$
- Bottom-Right (60) and Top-Left (12): $60 \times 12 = 720$
Products: 320, 20, 80, 720.
Is there a relationship between the products of Fig 1 and Fig 2?
Fig 1 Products: 2, 4, 6, 18
Fig 2 Products: 320, 20, 80, 720
Let's check ratios:
$320 / 2 = 160$
$20 / 4 = 5$
$80 / 6 = 13.33$
$720 / 18 = 40$
No.
Let's check sums of opposite pairs again.
Fig 1 Sums:
$2+1=3$
$2+2=4$
$3+2=5$
$3+6=9$
Sums: 3, 4, 5, 9.
Fig 2 Sums:
$8+40=48$
$10+2=12$
$20+4=24$
$60+12=72$
Sums: 48, 12, 24, 72.
Look at the Fig 2 sums: 12, 24, 48, 72.
$12 \times 1 = 12$
$12 \times 2 = 24$
$12 \times 4 = 48$
$12 \times 6 = 72$
Multipliers: 1, 2, 4, 6.
Now look at Fig 1 sums: 3, 4, 5, 9.
Do these match the multipliers 1, 2, 4, 6? No.
But wait.
Pair (Top-Right/Bottom-Left) in Fig 2 had Sum 12. Multiplier 1.
Pair (Right/Left) in Fig 2 had Sum 24. Multiplier 2.
Pair (Top/Bottom) in Fig 2 had Sum 48. Multiplier 4.
Pair (Bottom-Right/Top-Left) in Fig 2 had Sum 72. Multiplier 6.
Let's see if the Fig 1 sums correspond to these positions.
Position: Top-Right/Bottom-Left. Fig 1 Sum = $2+2=4$.
Position: Right/Left. Fig 1 Sum = $3+2=5$.
Position: Top/Bottom. Fig 1 Sum = $2+1=3$.
Position: Bottom-Right/Top-Left. Fig 1 Sum = $3+6=9$.
Sums: 4, 5, 3, 9.
Multipliers needed for Fig 2 were: 1, 2, 4, 6.
No direct match.
However, look at the products of the Fig 1 sums?
$3 \times 4 \times 5 \times 9 = 540$.
$12 \times 24 \times 48 \times 72 = \dots$ huge.
Let's step back. What is the most common simple logic for these "find the center number" puzzles?
1. Sum of all outer numbers.
2. Sum of opposite pairs is constant.
3. (Sum of some) - (Sum of others) = Center.
4. Product of some / Product of others.
Let's try (Sum of Even Positions) - (Sum of Odd Positions).
Fig 2:
Even pos (12, 3, 6, 9 o'clock approx?): 8, 20, 40, 4. Sum = 72.
Odd pos (1:30, 4:30, 7:30, 10:30): 10, 60, 2, 12. Sum = 84.
Diff = 12.
Fig 1:
Even pos (Top, Right, Bottom, Left): 2, 3, 1, 2. Sum = 8.
Odd pos (TR, BR, BL, TL): 2, 3, 2, 6. Sum = 13.
Diff = $13 - 8 = 5$.
So, if the rule is "Difference between the sum of diagonal/intercardinal numbers and cardinal numbers", then:
Fig 2 Center would be 12 (or -12).
Fig 1 Center would be 5 (or -5).
Is it possible the second image *has* a center number that I'm supposed to infer, or is the "12" in the difference significant?
Wait, look at the numbers in Fig 2 again.
12, 24, 48, 72 are the sums of opposite pairs.
$12 = 12 \times 1$
$24 = 12 \times 2$
$48 = 12 \times 4$
$72 = 12 \times 6$
The multipliers are 1, 2, 4, 6.
Sum of multipliers = $1+2+4+6 = 13$.
Product? $1 \times 2 \times 4 \times 6 = 48$.
Now look at Fig 1 opposite sums: 3, 4, 5, 9.
Is there a base number here?
GCD of 3,4,5,9 is 1.
Not helpful.
Let's try another very common pattern: Center = Sum of Outer Numbers / 2 or similar.
Fig 2 Sum = 156. $156 / ? = $ Integer?
$156 / 12 = 13$.
$156 / 13 = 12$.
Fig 1 Sum = 21.
$21 / ? = $ Integer?
$21 / 3 = 7$.
$21 / 7 = 3$.
Let's look at the relationship between the two figures again.
Maybe the center of Fig 1 is related to the center of Fig 2?
If the rule is consistent, and we assume the "Difference of alternating sums" rule:
Fig 2 Diff = 12.
Fig 1 Diff = 5.
Is "5" a plausible answer?
Let's check if there's a simpler additive rule.
Fig 2: $8+10+20+60+40+2+4+12 = 156$.
Fig 1: $6+2+2+3+3+1+2+2 = 21$.
What if the center is the sum of the digits?
Fig 2: $1+5+6 = 12$.
Fig 1: $2+1 = 3$.
What if the center is the number of letters in the English word for the sum? No.
Let's look at the visual arrangement of Fig 1 again.
It looks like the numbers are grouped.
Top half: 6, 2, 2. Sum = 10.
Bottom half: 2, 1, 3, 3, 2? No.
Left half: 6, 2, 2, 1. Sum = 11.
Right half: 2, 3, 3, 2. Sum = 10.
Diff = 1.
Top half (above horizontal midline): 6, 2, 2. (Assuming 6 is TL, 2 is T, 2 is TR). Sum = 10.
Bottom half: 2, 1, 3, 3? (BL, B, BR, R??).
Let's define halves strictly.
Horizontal split:
Top: 6 (TL), 2 (T), 2 (TR). Sum = 10.
Bottom: 2 (L), 1 (B), 3 (BR), 3 (R)? No, L and R are on the midline.
Usually "Top" includes Top-Left, Top, Top-Right.
"Bottom" includes Bottom-Left, Bottom, Bottom-Right.
What about Left and Right?
If we do Vertical Split:
Left: 6 (TL), 2 (L), 2 (BL). Sum = 10.
Right: 2 (TR), 3 (R), 3 (BR). Sum = 8.
Top: 6 (TL), 2 (T), 2 (TR). Sum = 10.
Bottom: 2 (BL), 1 (B), 3 (BR). Sum = 6.
Left+Right = 18.
Top+Bottom = 16.
Remaining numbers: L(2) and R(3).
This is getting messy.
Let's go back to the Opposite Pairs Sum idea for Fig 2.
Sums: 12, 24, 48, 72.
These are all multiples of 12.
$1 \times 12, 2 \times 12, 4 \times 12, 6 \times 12$.
The coefficients are 1, 2, 4, 6.
Note that $1+2+4+6 = 13$.
Also note that the numbers in the "1" coefficient pair are 10 and 2.
"2" coeff pair: 20 and 4.
"4" coeff pair: 8 and 40.
"6" coeff pair: 60 and 12.
Is there a pattern in the coefficients relative to position?
Pos 1:30 (10,2) -> Coeff 1.
Pos 3:00 (20,4) -> Coeff 2.
Pos 12:00 (8,40) -> Coeff 4.
Pos 4:30 (60,12) -> Coeff 6.
Now apply this "Coefficient" logic to Fig 1?
We need to find a "Base Number" for Fig 1 such that the opposite sums are multiples of it.
Fig 1 Opposite Sums: 3, 4, 5, 9.
GCD(3,4,5,9) = 1.
So Base = 1.
Coefficients: 3, 4, 5, 9.
Sum of Coefficients = $3+4+5+9 = 21$.
Does the center equal the sum of coefficients?
If so, Fig 2 Center = 13.
Fig 1 Center = 21.
Let's check if 13 makes sense for Fig 2.
Sum of outer numbers = 156.
$156 / 12 = 13$.
So, Center = Sum of Outer Numbers / Base Multiple.
For Fig 2: Base Multiple was 12. Sum = 156. Center = 13.
For Fig 1: Base Multiple was 1. Sum = 21. Center = 21.
Alternatively, Center = Sum of Coefficients.
Fig 2 Coeffs: 1,2,4,6. Sum = 13.
Fig 1 Coeffs: 3,4,5,9. Sum = 21.
This seems like a strong candidate. The logic is:
1. Calculate sum of opposite pairs.
2. Find the Greatest Common Divisor (GCD) of these sums. This is the "Base".
3. Divide each pair sum by the Base to get "Coefficients".
4. The Center Number is the Sum of these Coefficients.
Let's double check this logic.
Why would the center be the sum of coefficients?
In Fig 2, Sum of Outer = 156.
Sum of Pair Sums = $12+24+48+72 = 156$.
Sum of Pair Sums = Base $\times$ (Sum of Coeffs).
$156 = 12 \times (\text{Sum of Coeffs})$.
$\text{Sum of Coeffs} = 156 / 12 = 13$.
So, Center = Sum of Outer Numbers / GCD of Opposite Pair Sums.
Let's apply this formula to Fig 1.
1. Opposite Pair Sums: 3, 4, 5, 9.
2. GCD(3, 4, 5, 9) = 1.
3. Sum of Outer Numbers = 21.
4. Center = $21 / 1 = 21$.
Let's try one more hypothesis. What if the center is simply the Sum of the Outer Numbers for Fig 1, because the GCD is 1?
And for Fig 2, the center is 13?
Is there any other interpretation?
What if the rule is simpler?
Look at Fig 2:
$8+10+20+60+40+2+4+12 = 156$.
$1+5+6=12$? No.
$156 / 12 = 13$.
Look at Fig 1:
$6+2+2+3+3+1+2+2 = 21$.
$21 / 1 = 21$.
Is it possible the answer is just the sum?
If the GCD was always 1, yes. But in Fig 2, the GCD is 12.
Why divide by GCD?
Maybe the rule is: Center = Sum of Opposite Pair Sums / Largest Pair Sum?
Fig 2: $156 / 72 = 2.16$. No.
Center = Sum of Opposite Pair Sums / Smallest Pair Sum?
Fig 2: $156 / 12 = 13$.
Fig 1: $21 / 3 = 7$.
Let's compare "Divide by GCD" vs "Divide by Smallest Sum".
In Fig 2, GCD=12, Smallest Sum=12. They are the same.
In Fig 1, GCD=1, Smallest Sum=3.
If rule is "Divide by Smallest Sum": Answer = 7.
If rule is "Divide by GCD": Answer = 21.
Which is more "standard"?
Usually, these puzzles rely on integer properties.
GCD is a more robust mathematical property than "Smallest Sum".
However, 21 is the total sum. If the answer is just the total sum, why the complex structure in Fig 2?
In Fig 2, if the answer was the total sum, it would be 156.
If the answer is 13, it's a derived number.
Let's look for clues in the numbers themselves.
Fig 1: 1, 2, 2, 2, 2, 3, 3, 6.
Fig 2: 2, 4, 8, 10, 12, 20, 40, 60.
Notice Fig 2 numbers are all even.
Fig 1 has odd numbers (1, 3, 3).
Let's try Average of Opposite Sums.
Fig 2: $(12+24+48+72)/4 = 156/4 = 39$.
Fig 1: $(3+4+5+9)/4 = 21/4 = 5.25$. No.
Let's try Sum of Unique Digits?
No.
Let's reconsider the Difference of Alternating Sums.
Fig 2: $|84 - 72| = 12$.
Fig 1: $|13 - 8| = 5$.
Between 21, 7, and 5, which is the best fit?
The "GCD" method yielded 21.
The "Smallest Sum" method yielded 7.
The "Alternating Difference" method yielded 5.
Let's look at the shape again.
Fig 1 is a square/diamond.
Fig 2 is a star.
Often, the number of points matters. Both have 8.
Let's try one more calculation for Fig 2.
$(8 \times 40) / (10 \times 2) = 320 / 20 = 16$.
$(60 \times 12) / (20 \times 4) = 720 / 80 = 9$.
$16 - 9 = 7$?
$16 + 9 = 25$?
Let's try for Fig 1.
$(2 \times 1) / (2 \times 2) = 2 / 4 = 0.5$.
$(3 \times 6) / (3 \times 2) = 18 / 6 = 3$.
$3 - 0.5 = 2.5$.
$3 + 0.5 = 3.5$.
No.
Let's go with the most mathematically elegant solution found:
Rule: The center number is the sum of the outer numbers divided by the greatest common divisor (GCD) of the sums of opposite pairs.
Verification for Figure 2:
- Opposite sums: 12, 24, 48, 72.
- GCD(12, 24, 48, 72) = 12.
- Total Sum = 156.
- Center = $156 / 12 = 13$.
Application to Figure 1:
- Opposite sums: 3, 4, 5, 9.
- GCD(3, 4, 5, 9) = 1.
- Total Sum = 21.
- Center = $21 / 1 = 21$.
Alternative Check:
Is there a simpler rule?
Sum of outer numbers for Fig 1 is 21.
Sum of outer numbers for Fig 2 is 156.
If the answer for Fig 2 was 156, the rule would be "Sum".
But 156 is a large number. 13 is "cleaner".
If the answer for Fig 1 is 21, it is just the sum.
Let's consider if "5" (from alternating difference) is better.
Fig 2 Diff = 12.
Fig 1 Diff = 5.
Is 12 a "special" number in Fig 2? Yes, it's the GCD and the smallest pair sum.
Is 5 a "special" number in Fig 1?
Smallest pair sum is 3. GCD is 1.
The alternating difference is 5.
Comparing "13" vs "12" for Fig 2.
If the center of Fig 2 was 12, the rule would be "Alternating Difference".
If the center of Fig 2 was 13, the rule would be "Sum / GCD".
Without the center of Fig 2 provided, we have to guess the intended complexity.
"Alternating Difference" is a very common puzzle trope.
"Sum / GCD" is rare.
Let's look at the numbers in Fig 1 again.
6, 2, 2, 3, 3, 1, 2, 2.
Alternating Sum 1: $6+2+3+2 = 13$.
Alternating Sum 2: $2+3+1+2 = 8$.
Diff = 5.
Let's look at Fig 2 again.
Alternating Sum 1: $12+10+60+2 = 84$. (Starting TL, going clockwise: TL, TR, BR, BL? No, TL, TR, BR, BL are not alternating indices. Indices 1,3,5,7 vs 2,4,6,8).
Let's use indices 1-8 clockwise from Top.
1:8, 2:10, 3:20, 4:60, 5:40, 6:2, 7:4, 8:12.
Odd indices (1,3,5,7): $8+20+40+4 = 72$.
Even indices (2,4,6,8): $10+60+2+12 = 84$.
Diff = 12.
If the rule is Absolute Difference of Alternating Sums, then:
Fig 2 Center = 12.
Fig 1 Center = 5.
This feels like a very standard IQ test pattern. The "Sum/GCD" is too complex. The "Alternating Difference" is simple and uses basic arithmetic.
Final Answer Prediction: 5.
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Final Answer:
5
Parent Tip: Review the logic above to help your child master the concept of free maths puzzle worksheet.