How I figured it out:
This puzzle has a grid of 4 boxes in a square! Each box has 2 circles next to
it, and it also touches 2 other boxes.
Let's give each box a letter name so it is easy to track:
• A is the top-left box.
• B is the top-right box.
• C is the bottom-left box.
• D is the bottom-right box.
My Equations for each box:
For box A: The numbers around it are circles 7 and 2, plus boxes B and C.
4 × A = 7 + 2 + B + C
4 × A = 9 + B + C — (Equation 1)
For box B: The numbers around it are circles 7 and 4, plus boxes A and D.
4 × B = 7 + 4 + A + D
4 × B = 11 + A + D — (Equation 2)
For box C: The numbers around it are circles 9 and 6, plus boxes A and D.
4 × C = 9 + 6 + A + D
4 × C = 15 + A + D — (Equation 3)
For box D: The numbers around it are circles 10 and 19, plus boxes B and C.
4 × D = 10 + 19 + B + C
4 × D = 29 + B + C — (Equation 4)
Finding the numbers:
First, I notice that the right side of Equation 1 and Equation 4 both use (B +
C)!
4 × D = 29 + B + C
Since 9 + B + C = 4 × A, then B + C = (4 × A) - 9.
So, 4 × D = 29 + (4 × A) - 9
4 × D = 20 + 4 × A
Divide everything by 4: D = 5 + A
Next, I compare Equation 2 and Equation 3. Both use (A + D)!
4 × C = 15 + A + D
4 × B = 11 + A + D
If we subtract them: (4 × C) - (4 × B) = 15 - 11 = 4.
Divide by 4: C - B = 1, which means C = B + 1.
Now let's put D = 5 + A into Equation 2:
4 × B = 11 + A + (5 + A)
4 × B = 16 + 2 × A
Divide by 2: 2 × B = 8 + A, so A = 2 × B - 8.
Now let's use Equation 1 with our new clues (C = B + 1 and A = 2 × B - 8):
4 × (2 × B - 8) = 9 + B + (B + 1)
8 × B - 32 = 10 + 2 × B
Move the B's to one side and numbers to the other:
6 × B = 42
B = 42 ÷ 6
B = 7
Now I can find all the other boxes easily!
• C = B + 1 = 7 + 1 → C = 8
• A = 2 × 7 - 8 = 14 - 8 → A = 6
• D = 5 + A = 5 + 6 → D = 11
My Final Answers:
Top-Left Box (A) = 6
Top-Right Box (B) = 7
Bottom-Left Box (C) = 8
Bottom-Right Box (D) = 11
Checking my answers:
• Top-Left Box: (7 + 2 + 7 + 8) ÷ 4 = 24 ÷ 4 = 6. (Works!)
• Top-Right Box: (7 + 4 + 6 + 11) ÷ 4 = 28 ÷ 4 = 7. (Works!)
• Bottom-Left Box: (9 + 6 + 6 + 11) ÷ 4 = 32 ÷ 4 = 8. (Works!)
• Bottom-Right Box: (10 + 19 + 7 + 8) ÷ 4 = 44 ÷ 4 = 11. (Works!)