Each circle, lettered A through H, has its own number value from 1 to 8.
No two circles have the same value.
The numbers shown in the diagram are the sums of the circles that overlap at those points.
For example, 9 is the sum of circles B and C.
Can you determine the value of each circle?
Solution:
The answers are:
A = 6
B = 1
C = 8
D = 4
E = 7
F = 3
G = 2
H = 5
First set up the seven equations from the diagram:
B + C = 9 (1)
A + C + F = 17 (2)
A + F + H = 14 (3)
A + G + G = 13 (4)
A + D + G = 12 (5)
D + E = 11 (6)
H + G = 7 (7)
I then subtracted several equations from another to get:
C - H = 3 or C = H + 3 (2) - (3), call this equation (8)
H - D = 1 or H = D + 1 (4) - (5), call this equation (9)
F - G = 1 or F = G + 1 (3) - (4), call this equation (10)
I rewrote three of the equations (#1, #6, and #7):
C = 9 - B (1), call this equation (11)
D = 11 - E (6), call this equation (12)
H = 7 - G (7), call this equation (13)
There are eight variables and only seven equations, so there would be an infinite number of solutions if we didn't have the additional clue that each variable must be a different number between one and eight.
Take one of the variables, and substitute the numbers 1 to 8 for it until you find values that work for all eight variables.
I began with the variable H.
If H = 1, then D = 0 by equation #9, which is not allowed.
If H = 2, then D = 1 (equation #9) and E = 10 (equation #12), which is not allowed
If H = 3, then D = 2 (equation #9) and E = 9 (equation #12), which is not allowed
If H = 4, then D = 3 (equation #9), E = 8 (equation #12), C = 7 (equation #8), G = 3 (equation #13), which is not allowed (D = G = 3)
If H = 5, then C = 8 (equation #8), D = 4 (equation #9), E = 7 (equation #11), B = 1 (equation #10), G = 2 (equation #13), F = 3 (equation #10), and A = 6 (equation #2, #3, #4, or #5)
These eight values work in all seven of the original equations, so that is the solution.
Correctly solved by:
| 1. K. Sengupta | Calcutta, India |
| 2. Kamal Lohia |
Holy Angel School, Hisar, Haryana, India |
| 3. Ivy Joseph | Pune, Maharashtra, India |
| 4. Colin (Yowie) Bowey | Beechworth, Victoria, Australia |
| 5. Dr. Hari Kishan |
D.N. College, Meerut, Uttar Pradesh, India |
| 6. Davit Banana | Istanbul, Turkey |
| 7. Kelly Stubblefield | Mobile, Alabama, USA |