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Step-by-step solution for: Use the same rule to fill the hexagons below. Now you also make ...
It looks like you've uploaded an image of a circular puzzle with numbers and shapes arranged around a central circle. Based on the description and typical patterns in such puzzles, this appears to be a number sequence or logic puzzle where numbers are placed around a circle, connected by shapes (squares and circles), and there may be a pattern or rule that links them.
Let me analyze the visible elements:
- Numbers: 9, 108, 7, 17, 6, 11
- Shapes: Squares and Circles alternating around the circle.
- The numbers are placed in circles, and squares are between them.
- Some squares are colored differently (green, pink, blue, yellow, orange).
- There is a number 108 next to 9, which suggests a possible multiplication or transformation rule.
Let’s map the positions clockwise:
1. Circle: 9
2. Square: (empty)
3. Circle: 108
4. Square: (empty)
5. Circle: 7
6. Square: (empty)
7. Circle: 17
8. Square: (empty)
9. Circle: 6
10. Square: (empty)
11. Circle: 11
12. Square: (empty)
Wait — actually, there are 6 numbers and 6 squares. So it's a cycle of 12 elements: alternating circle and square.
But let's list them in order around the circle:
Starting from top:
1. 9 (circle) → green square → 108 (circle) → pink square → 7 (circle) → orange square → 17 (circle) → yellow square → 6 (circle) → blue square → 11 (circle) → pink square? Wait — maybe not all colors are unique.
Actually, looking at the structure:
It seems like:
- Each number is in a circle.
- Between each pair of circles is a square.
- There are 6 numbers and 6 squares.
So the sequence is:
- 9 → [square] → 108 → [square] → 7 → [square] → 17 → [square] → 6 → [square] → 11 → [square] → back to 9
Now, we need to find a pattern or rule connecting the numbers, possibly through the squares.
Let’s look at adjacent pairs:
1. 9 → 108
What operation turns 9 into 108?
9 × 12 = 108 → so maybe ×12?
2. 108 → 7
108 → 7? That doesn’t make sense unless it's modulo or some other rule.
Alternatively, maybe the squares represent operations or values that connect two numbers.
But perhaps the squares contain values that are derived from the adjacent numbers.
Wait — maybe the squares are meant to be filled with numbers based on a rule involving the adjacent circles.
But the problem says "solve the problem" — likely, the squares are missing numbers, and we need to find them.
Let’s assume that the squares are placeholders for numbers, and the pattern involves a mathematical relationship between adjacent numbers via the squares.
Another idea: Maybe the product or sum of adjacent numbers relates to the square?
Try checking:
- 9 and 108: 9 × 12 = 108 → could mean the square between them is 12?
- Then 108 and 7: what connects them? 108 ÷ 15.428... ≈ 7 → not nice.
- Or maybe the square represents a value that when combined with one number gives the next.
Let’s suppose the pattern is:
Number → Square → Number
And the square is a multiplier or operator.
Try:
From 9 to 108: 9 × 12 = 108 → so the square between 9 and 108 might be 12.
Then from 108 to 7: ?
If we do 108 / x = 7 → x = 108/7 ≈ 15.428 — not integer.
Alternatively, maybe the square is the result of an operation between the two numbers?
Try:
Is there a pattern like:
- Square = (left number + right number) / something?
Try averaging: (9 + 108)/2 = 58.5 — no.
Or maybe the squares are part of a sequence themselves?
List the numbers: 9, 108, 7, 17, 6, 11
Wait — perhaps these numbers are related to the positions or shapes?
Another idea: Maybe the sum of opposite numbers is constant?
Let’s try pairing opposite numbers.
Assuming the circle has 6 numbers, evenly spaced, so opposite pairs:
- 9 and 17 → 9 + 17 = 26
- 108 and 6 → 108 + 6 = 114
- 7 and 11 → 7 + 11 = 18 → not constant.
No.
What if we look at ratios?
- 108 / 9 = 12
- 17 / 7 ≈ 2.428
- 11 / 6 ≈ 1.833 — not helpful.
Wait — perhaps the squares are meant to be filled with numbers, and the rule is that the product of adjacent numbers equals something?
But we don't have any numbers in squares.
Alternatively, maybe the square between two numbers contains the result of a function of the two numbers.
But we need more info.
Wait — another possibility: This is a clockwise pattern where each number is derived from the previous via a rule involving the square.
But the square is between them — so perhaps the square contains the operation or the multiplier.
Let’s suppose the pattern is:
Left number × Square = Right number
Try:
1. 9 × ? = 108 → ? = 12 → so square between 9 and 108 = 12
2. 108 × ? = 7 → ? = 7/108 ≈ 0.0648 — not nice
3. 7 × ? = 17 → ? = 17/7 ≈ 2.428 — no
4. 17 × ? = 6 → ? = 6/17 ≈ 0.35 — no
5. 6 × ? = 11 → ? = 11/6 ≈ 1.83 — no
6. 11 × ? = 9 → ? = 9/11 ≈ 0.818 — no
Not working.
Reverse: maybe Right number × Square = Left number?
1. 108 × ? = 9 → ? = 9/108 = 1/12 — possible, but then square would be 1/12 — unlikely.
Alternatively, maybe addition?
9 + ? = 108 → ? = 99 — possible.
Then 108 + ? = 7 → ? = -101 — possible, but inconsistent.
Not promising.
Another idea: Perhaps the squares are not numbers, but the colors matter? But no indication.
Wait — look at the numbers: 9, 108, 7, 17, 6, 11
Let’s see if they follow a pattern.
List them in order: 9, 108, 7, 17, 6, 11
No obvious arithmetic progression.
But notice: 9 and 108 — 108 = 9 × 12
Then 7 and 17 — difference 10
6 and 11 — difference 5
Not clear.
Wait — maybe the squares are supposed to be filled with numbers such that the sum of a number and the next number is constant?
9 + 108 = 117
108 + 7 = 115
7 + 17 = 24
17 + 6 = 23
6 + 11 = 17
11 + 9 = 20 — not constant.
No.
Another idea: Maybe the product of opposite numbers is constant?
Opposite pairs:
- 9 and 17 → 9×17 = 153
- 108 and 6 → 108×6 = 648
- 7 and 11 → 7×11 = 77 — not constant.
No.
Wait — perhaps the squares are meant to be filled with numbers that are the average or difference?
But without knowing the rule, it's hard.
Let’s think differently.
Maybe the number in the square is the sum of the two adjacent numbers?
Try:
- Between 9 and 108: 9 + 108 = 117 → put 117 in square?
- Between 108 and 7: 108 + 7 = 115
- Between 7 and 17: 7 + 17 = 24
- Between 17 and 6: 17 + 6 = 23
- Between 6 and 11: 6 + 11 = 17
- Between 11 and 9: 11 + 9 = 20
But then the squares would be: 117, 115, 24, 23, 17, 20 — but no pattern.
Alternatively, maybe the square is the product?
9×108 = 972 — too big.
No.
Wait — look at the color coding:
- Green square between 9 and 108
- Pink between 108 and 7
- Orange between 7 and 17
- Yellow between 17 and 6
- Blue between 6 and 11
- Another pink between 11 and 9
Pink appears twice — maybe not significant.
But perhaps the value in the square is determined by the numbers.
Let’s go back to the first pair: 9 to 108
9 × 12 = 108
Now, 12 is a nice number.
Next: 108 to 7 — can we get from 108 to 7 via division? 108 / 15.428... = 7 — no.
But 108 - 101 = 7 — so difference of 101.
Not helpful.
Wait — maybe the square is not a number, but the operation? But the puzzle likely wants us to fill in the squares with numbers.
Another idea: Perhaps the sum of all numbers or products relate.
Sum: 9 + 108 + 7 + 17 + 6 + 11 = 158
Not helpful.
Wait — here's a new idea:
Maybe the number in the circle and the number in the square are related to the next circle via a fixed rule.
For example, perhaps:
Square = (Right number) / (Left number)
Try:
- Between 9 and 108: 108 / 9 = 12 → so square = 12
- Between 108 and 7: 7 / 108 ≈ 0.0648 — not nice
- Between 7 and 17: 17 / 7 ≈ 2.428 — no
- Between 17 and 6: 6 / 17 ≈ 0.35 — no
- Between 6 and 11: 11 / 6 ≈ 1.83 — no
- Between 11 and 9: 9 / 11 ≈ 0.818 — no
Only first one works.
But wait — what if the rule is multiplication by a constant?
9 → 108: ×12
108 → 7: not × anything nice
Unless it's not sequential.
Another possibility: The squares are the results of an operation, and the numbers are inputs.
But we need a different approach.
Let’s consider that the shape (square) might represent a function.
Perhaps the puzzle is about prime factors or digit sums.
Check digit sums:
- 9: 9
- 108: 1+0+8=9
- 7: 7
- 17: 1+7=8
- 6: 6
- 11: 1+1=2
No pattern.
Digit sum of 9 and 108 both 9 — interesting.
Then 7 and 17: 7 and 8 — close.
6 and 11: 6 and 2 — not.
Wait — 9 and 108: both have digit sum 9
7 and 17: 7 and 8 — not same
6 and 11: 6 and 2 — no
But 108 has digit sum 9, and 9 is the previous number — maybe the digit sum of the next number is the current number?
108 has digit sum 9 — matches previous number 9.
Then next number is 7 — does 7 have digit sum 7? Yes, but previous was 108 — digit sum 9, not 7.
Wait — reverse: the current number is the digit sum of the next number?
- 9 → next is 108 → digit sum 1+0+8=9 → yes!
- 108 → next is 7 → digit sum of 7 is 7 → but 108 ≠ 7 → no
- 7 → next is 17 → digit sum of 17 is 8 ≠ 7
- 17 → next is 6 → digit sum of 6 is 6 ≠ 17
- 6 → next is 11 → digit sum of 11 is 2 ≠ 6
- 11 → next is 9 → digit sum of 9 is 9 ≠ 11
Only first one works.
But wait — what if the square contains the digit sum of the next number?
- Between 9 and 108: digit sum of 108 is 9 → so square = 9
- Between 108 and 7: digit sum of 7 is 7 → square = 7
- Between 7 and 17: digit sum of 17 is 8 → square = 8
- Between 17 and 6: digit sum of 6 is 6 → square = 6
- Between 6 and 11: digit sum of 11 is 2 → square = 2
- Between 11 and 9: digit sum of 9 is 9 → square = 9
So squares would be: 9, 7, 8, 6, 2, 9
But is that consistent?
But the square between 9 and 108 is green — maybe it's supposed to be 9.
But we already have 9 in the circle — maybe duplicate is ok.
But let's check if this makes sense.
The rule: The square between A and B contains the digit sum of B
So:
- After 9 comes 108 → digit sum of 108 is 9 → square = 9
- After 108 comes 7 → digit sum of 7 is 7 → square = 7
- After 7 comes 17 → digit sum of 17 is 8 → square = 8
- After 17 comes 6 → digit sum of 6 is 6 → square = 6
- After 6 comes 11 → digit sum of 11 is 2 → square = 2
- After 11 comes 9 → digit sum of 9 is 9 → square = 9
So the squares should be: 9, 7, 8, 6, 2, 9
But the puzzle likely wants us to fill in the squares with numbers.
And this rule explains why 108 is after 9: because its digit sum is 9.
Similarly, 7 is after 108 — digit sum of 7 is 7, but that doesn't explain why it's next.
But the rule is only about the square — not why the numbers are in that order.
Perhaps the numbers are arranged in a way that the digit sum of each number is the value in the square before it.
Wait — let's try:
- Before 9: the square is between 11 and 9 → digit sum of 9 is 9 → so square = 9
- Before 108: square between 9 and 108 → digit sum of 108 is 9 → square = 9
- Before 7: square between 108 and 7 → digit sum of 7 is 7 → square = 7
- Before 17: square between 7 and 17 → digit sum of 17 is 8 → square = 8
- Before 6: square between 17 and 6 → digit sum of 6 is 6 → square = 6
- Before 11: square between 6 and 11 → digit sum of 11 is 2 → square = 2
So the squares are:
- Between 9 and 108: 9
- Between 108 and 7: 7
- Between 7 and 17: 8
- Between 17 and 6: 6
- Between 6 and 11: 2
- Between 11 and 9: 9
But the square between 11 and 9 is also 9 — so two 9s.
But is there a better rule?
Wait — what if the square contains the digit sum of the number in the circle?
For example:
- Circle: 9 → digit sum = 9 → so square before it = 9
- Circle: 108 → digit sum = 9 → so square before it = 9
- Circle: 7 → digit sum = 7 → square before it = 7
- Circle: 17 → digit sum = 8 → square before it = 8
- Circle: 6 → digit sum = 6 → square before it = 6
- Circle: 11 → digit sum = 2 → square before it = 2
So the square immediately before each circle contains the digit sum of that circle.
That means:
- Before 9: square = 9
- Before 108: square = 9
- Before 7: square = 7
- Before 17: square = 8
- Before 6: square = 6
- Before 11: square = 2
Now, let's assign the squares in order:
Starting from the square between 9 and 108: this is the square before 108, so it should be 9.
Then square between 108 and 7: before 7 → should be 7.
Then between 7 and 17: before 17 → 8
Between 17 and 6: before 6 → 6
Between 6 and 11: before 11 → 2
Between 11 and 9: before 9 → 9
So the squares are:
- Green (between 9 and 108): 9
- Pink (between 108 and 7): 7
- Orange (between 7 and 17): 8
- Yellow (between 17 and 6): 6
- Blue (between 6 and 11): 2
- Last pink (between 11 and 9): 9
This fits perfectly!
Moreover, the number 108 has digit sum 9, and it follows 9, which is consistent.
Also, 11 has digit sum 2, and it follows 6, and the square before it is 2.
So the rule is:
> The number in the square immediately preceding a circle is equal to the digit sum of the number in that circle.
Therefore, the solution is to fill in the squares with:
- Green square (before 108): 9
- Pink square (before 7): 7
- Orange square (before 17): 8
- Yellow square (before 6): 6
- Blue square (before 11): 2
- Final pink square (before 9): 9
| Square Color | Value |
|--------------|-------|
| Green | 9 |
| Pink | 7 |
| Orange | 8 |
| Yellow | 6 |
| Blue | 2 |
| Pink (last) | 9 |
Thus, the missing numbers in the squares are: 9, 7, 8, 6, 2, 9 in clockwise order starting from the green square.
✔ Solution: Fill the squares with the digit sum of the number in the following circle.
Let me analyze the visible elements:
Observed Elements:
- Numbers: 9, 108, 7, 17, 6, 11
- Shapes: Squares and Circles alternating around the circle.
- The numbers are placed in circles, and squares are between them.
- Some squares are colored differently (green, pink, blue, yellow, orange).
- There is a number 108 next to 9, which suggests a possible multiplication or transformation rule.
Let’s map the positions clockwise:
1. Circle: 9
2. Square: (empty)
3. Circle: 108
4. Square: (empty)
5. Circle: 7
6. Square: (empty)
7. Circle: 17
8. Square: (empty)
9. Circle: 6
10. Square: (empty)
11. Circle: 11
12. Square: (empty)
Wait — actually, there are 6 numbers and 6 squares. So it's a cycle of 12 elements: alternating circle and square.
But let's list them in order around the circle:
Starting from top:
1. 9 (circle) → green square → 108 (circle) → pink square → 7 (circle) → orange square → 17 (circle) → yellow square → 6 (circle) → blue square → 11 (circle) → pink square? Wait — maybe not all colors are unique.
Actually, looking at the structure:
It seems like:
- Each number is in a circle.
- Between each pair of circles is a square.
- There are 6 numbers and 6 squares.
So the sequence is:
- 9 → [square] → 108 → [square] → 7 → [square] → 17 → [square] → 6 → [square] → 11 → [square] → back to 9
Now, we need to find a pattern or rule connecting the numbers, possibly through the squares.
Let’s look at adjacent pairs:
1. 9 → 108
What operation turns 9 into 108?
9 × 12 = 108 → so maybe ×12?
2. 108 → 7
108 → 7? That doesn’t make sense unless it's modulo or some other rule.
Alternatively, maybe the squares represent operations or values that connect two numbers.
But perhaps the squares contain values that are derived from the adjacent numbers.
Wait — maybe the squares are meant to be filled with numbers based on a rule involving the adjacent circles.
But the problem says "solve the problem" — likely, the squares are missing numbers, and we need to find them.
Let’s assume that the squares are placeholders for numbers, and the pattern involves a mathematical relationship between adjacent numbers via the squares.
Another idea: Maybe the product or sum of adjacent numbers relates to the square?
Try checking:
- 9 and 108: 9 × 12 = 108 → could mean the square between them is 12?
- Then 108 and 7: what connects them? 108 ÷ 15.428... ≈ 7 → not nice.
- Or maybe the square represents a value that when combined with one number gives the next.
Let’s suppose the pattern is:
Number → Square → Number
And the square is a multiplier or operator.
Try:
From 9 to 108: 9 × 12 = 108 → so the square between 9 and 108 might be 12.
Then from 108 to 7: ?
If we do 108 / x = 7 → x = 108/7 ≈ 15.428 — not integer.
Alternatively, maybe the square is the result of an operation between the two numbers?
Try:
Is there a pattern like:
- Square = (left number + right number) / something?
Try averaging: (9 + 108)/2 = 58.5 — no.
Or maybe the squares are part of a sequence themselves?
List the numbers: 9, 108, 7, 17, 6, 11
Wait — perhaps these numbers are related to the positions or shapes?
Another idea: Maybe the sum of opposite numbers is constant?
Let’s try pairing opposite numbers.
Assuming the circle has 6 numbers, evenly spaced, so opposite pairs:
- 9 and 17 → 9 + 17 = 26
- 108 and 6 → 108 + 6 = 114
- 7 and 11 → 7 + 11 = 18 → not constant.
No.
What if we look at ratios?
- 108 / 9 = 12
- 17 / 7 ≈ 2.428
- 11 / 6 ≈ 1.833 — not helpful.
Wait — perhaps the squares are meant to be filled with numbers, and the rule is that the product of adjacent numbers equals something?
But we don't have any numbers in squares.
Alternatively, maybe the square between two numbers contains the result of a function of the two numbers.
But we need more info.
Wait — another possibility: This is a clockwise pattern where each number is derived from the previous via a rule involving the square.
But the square is between them — so perhaps the square contains the operation or the multiplier.
Let’s suppose the pattern is:
Left number × Square = Right number
Try:
1. 9 × ? = 108 → ? = 12 → so square between 9 and 108 = 12
2. 108 × ? = 7 → ? = 7/108 ≈ 0.0648 — not nice
3. 7 × ? = 17 → ? = 17/7 ≈ 2.428 — no
4. 17 × ? = 6 → ? = 6/17 ≈ 0.35 — no
5. 6 × ? = 11 → ? = 11/6 ≈ 1.83 — no
6. 11 × ? = 9 → ? = 9/11 ≈ 0.818 — no
Not working.
Reverse: maybe Right number × Square = Left number?
1. 108 × ? = 9 → ? = 9/108 = 1/12 — possible, but then square would be 1/12 — unlikely.
Alternatively, maybe addition?
9 + ? = 108 → ? = 99 — possible.
Then 108 + ? = 7 → ? = -101 — possible, but inconsistent.
Not promising.
Another idea: Perhaps the squares are not numbers, but the colors matter? But no indication.
Wait — look at the numbers: 9, 108, 7, 17, 6, 11
Let’s see if they follow a pattern.
List them in order: 9, 108, 7, 17, 6, 11
No obvious arithmetic progression.
But notice: 9 and 108 — 108 = 9 × 12
Then 7 and 17 — difference 10
6 and 11 — difference 5
Not clear.
Wait — maybe the squares are supposed to be filled with numbers such that the sum of a number and the next number is constant?
9 + 108 = 117
108 + 7 = 115
7 + 17 = 24
17 + 6 = 23
6 + 11 = 17
11 + 9 = 20 — not constant.
No.
Another idea: Maybe the product of opposite numbers is constant?
Opposite pairs:
- 9 and 17 → 9×17 = 153
- 108 and 6 → 108×6 = 648
- 7 and 11 → 7×11 = 77 — not constant.
No.
Wait — perhaps the squares are meant to be filled with numbers that are the average or difference?
But without knowing the rule, it's hard.
Let’s think differently.
Maybe the number in the square is the sum of the two adjacent numbers?
Try:
- Between 9 and 108: 9 + 108 = 117 → put 117 in square?
- Between 108 and 7: 108 + 7 = 115
- Between 7 and 17: 7 + 17 = 24
- Between 17 and 6: 17 + 6 = 23
- Between 6 and 11: 6 + 11 = 17
- Between 11 and 9: 11 + 9 = 20
But then the squares would be: 117, 115, 24, 23, 17, 20 — but no pattern.
Alternatively, maybe the square is the product?
9×108 = 972 — too big.
No.
Wait — look at the color coding:
- Green square between 9 and 108
- Pink between 108 and 7
- Orange between 7 and 17
- Yellow between 17 and 6
- Blue between 6 and 11
- Another pink between 11 and 9
Pink appears twice — maybe not significant.
But perhaps the value in the square is determined by the numbers.
Let’s go back to the first pair: 9 to 108
9 × 12 = 108
Now, 12 is a nice number.
Next: 108 to 7 — can we get from 108 to 7 via division? 108 / 15.428... = 7 — no.
But 108 - 101 = 7 — so difference of 101.
Not helpful.
Wait — maybe the square is not a number, but the operation? But the puzzle likely wants us to fill in the squares with numbers.
Another idea: Perhaps the sum of all numbers or products relate.
Sum: 9 + 108 + 7 + 17 + 6 + 11 = 158
Not helpful.
Wait — here's a new idea:
Maybe the number in the circle and the number in the square are related to the next circle via a fixed rule.
For example, perhaps:
Square = (Right number) / (Left number)
Try:
- Between 9 and 108: 108 / 9 = 12 → so square = 12
- Between 108 and 7: 7 / 108 ≈ 0.0648 — not nice
- Between 7 and 17: 17 / 7 ≈ 2.428 — no
- Between 17 and 6: 6 / 17 ≈ 0.35 — no
- Between 6 and 11: 11 / 6 ≈ 1.83 — no
- Between 11 and 9: 9 / 11 ≈ 0.818 — no
Only first one works.
But wait — what if the rule is multiplication by a constant?
9 → 108: ×12
108 → 7: not × anything nice
Unless it's not sequential.
Another possibility: The squares are the results of an operation, and the numbers are inputs.
But we need a different approach.
Let’s consider that the shape (square) might represent a function.
Perhaps the puzzle is about prime factors or digit sums.
Check digit sums:
- 9: 9
- 108: 1+0+8=9
- 7: 7
- 17: 1+7=8
- 6: 6
- 11: 1+1=2
No pattern.
Digit sum of 9 and 108 both 9 — interesting.
Then 7 and 17: 7 and 8 — close.
6 and 11: 6 and 2 — not.
Wait — 9 and 108: both have digit sum 9
7 and 17: 7 and 8 — not same
6 and 11: 6 and 2 — no
But 108 has digit sum 9, and 9 is the previous number — maybe the digit sum of the next number is the current number?
108 has digit sum 9 — matches previous number 9.
Then next number is 7 — does 7 have digit sum 7? Yes, but previous was 108 — digit sum 9, not 7.
Wait — reverse: the current number is the digit sum of the next number?
- 9 → next is 108 → digit sum 1+0+8=9 → yes!
- 108 → next is 7 → digit sum of 7 is 7 → but 108 ≠ 7 → no
- 7 → next is 17 → digit sum of 17 is 8 ≠ 7
- 17 → next is 6 → digit sum of 6 is 6 ≠ 17
- 6 → next is 11 → digit sum of 11 is 2 ≠ 6
- 11 → next is 9 → digit sum of 9 is 9 ≠ 11
Only first one works.
But wait — what if the square contains the digit sum of the next number?
- Between 9 and 108: digit sum of 108 is 9 → so square = 9
- Between 108 and 7: digit sum of 7 is 7 → square = 7
- Between 7 and 17: digit sum of 17 is 8 → square = 8
- Between 17 and 6: digit sum of 6 is 6 → square = 6
- Between 6 and 11: digit sum of 11 is 2 → square = 2
- Between 11 and 9: digit sum of 9 is 9 → square = 9
So squares would be: 9, 7, 8, 6, 2, 9
But is that consistent?
But the square between 9 and 108 is green — maybe it's supposed to be 9.
But we already have 9 in the circle — maybe duplicate is ok.
But let's check if this makes sense.
The rule: The square between A and B contains the digit sum of B
So:
- After 9 comes 108 → digit sum of 108 is 9 → square = 9
- After 108 comes 7 → digit sum of 7 is 7 → square = 7
- After 7 comes 17 → digit sum of 17 is 8 → square = 8
- After 17 comes 6 → digit sum of 6 is 6 → square = 6
- After 6 comes 11 → digit sum of 11 is 2 → square = 2
- After 11 comes 9 → digit sum of 9 is 9 → square = 9
So the squares should be: 9, 7, 8, 6, 2, 9
But the puzzle likely wants us to fill in the squares with numbers.
And this rule explains why 108 is after 9: because its digit sum is 9.
Similarly, 7 is after 108 — digit sum of 7 is 7, but that doesn't explain why it's next.
But the rule is only about the square — not why the numbers are in that order.
Perhaps the numbers are arranged in a way that the digit sum of each number is the value in the square before it.
Wait — let's try:
- Before 9: the square is between 11 and 9 → digit sum of 9 is 9 → so square = 9
- Before 108: square between 9 and 108 → digit sum of 108 is 9 → square = 9
- Before 7: square between 108 and 7 → digit sum of 7 is 7 → square = 7
- Before 17: square between 7 and 17 → digit sum of 17 is 8 → square = 8
- Before 6: square between 17 and 6 → digit sum of 6 is 6 → square = 6
- Before 11: square between 6 and 11 → digit sum of 11 is 2 → square = 2
So the squares are:
- Between 9 and 108: 9
- Between 108 and 7: 7
- Between 7 and 17: 8
- Between 17 and 6: 6
- Between 6 and 11: 2
- Between 11 and 9: 9
But the square between 11 and 9 is also 9 — so two 9s.
But is there a better rule?
Wait — what if the square contains the digit sum of the number in the circle?
For example:
- Circle: 9 → digit sum = 9 → so square before it = 9
- Circle: 108 → digit sum = 9 → so square before it = 9
- Circle: 7 → digit sum = 7 → square before it = 7
- Circle: 17 → digit sum = 8 → square before it = 8
- Circle: 6 → digit sum = 6 → square before it = 6
- Circle: 11 → digit sum = 2 → square before it = 2
So the square immediately before each circle contains the digit sum of that circle.
That means:
- Before 9: square = 9
- Before 108: square = 9
- Before 7: square = 7
- Before 17: square = 8
- Before 6: square = 6
- Before 11: square = 2
Now, let's assign the squares in order:
Starting from the square between 9 and 108: this is the square before 108, so it should be 9.
Then square between 108 and 7: before 7 → should be 7.
Then between 7 and 17: before 17 → 8
Between 17 and 6: before 6 → 6
Between 6 and 11: before 11 → 2
Between 11 and 9: before 9 → 9
So the squares are:
- Green (between 9 and 108): 9
- Pink (between 108 and 7): 7
- Orange (between 7 and 17): 8
- Yellow (between 17 and 6): 6
- Blue (between 6 and 11): 2
- Last pink (between 11 and 9): 9
This fits perfectly!
Moreover, the number 108 has digit sum 9, and it follows 9, which is consistent.
Also, 11 has digit sum 2, and it follows 6, and the square before it is 2.
So the rule is:
> The number in the square immediately preceding a circle is equal to the digit sum of the number in that circle.
Therefore, the solution is to fill in the squares with:
- Green square (before 108): 9
- Pink square (before 7): 7
- Orange square (before 17): 8
- Yellow square (before 6): 6
- Blue square (before 11): 2
- Final pink square (before 9): 9
Final Answer:
| Square Color | Value |
|--------------|-------|
| Green | 9 |
| Pink | 7 |
| Orange | 8 |
| Yellow | 6 |
| Blue | 2 |
| Pink (last) | 9 |
Thus, the missing numbers in the squares are: 9, 7, 8, 6, 2, 9 in clockwise order starting from the green square.
✔ Solution: Fill the squares with the digit sum of the number in the following circle.
Parent Tip: Review the logic above to help your child master the concept of magic hexagon worksheet.