+---+ +---+
| 3 |----------------| 6 |
+---+ +---+
/ | \ / |
/ | \ D / |
/ | \ / |
/ | \ / |
/ | \ / |
+---+ | +---+ |
| 1 | A | B | 4 | E `|
+---+ | +---+ |
\ | / \ |
\ | / \ |
\ | / \ |
\ | / C \ |
\ | / \ |
\ | / \ |
+---+ +---+
| 2 |---------------| 5 |
+---+ +---+

The diagram above represents the layout of a new prison.

- There are 21 guards, 45 prisoners, six towers for the guards and five compounds for the prisoners.
- The towers are numbered from 1 to 6 and the compounds lettered A to E.
- All of the compounds are triangular in shape and have a guard tower in each of the three corners.
- The number of prisoners in a compound is equal to the total number of guards in the towers on the corners of the compound.
- There are 11 prisoners in compound A, one guard in tower number 2, four guards in tower number 3.
- At least one guard must be in each tower.
- No two towers contain the same number of guards.
- No two compounds contain the same number of prisoners.

Based on the clues given above, determine the number of guards in each of the six towers?

Let Tower 1 = T1, Tower 2 = T2, etc.

a) T1 = A - T2 - T3 = 11 - 4 - 1 = 6

b) If the six towers add to 21, they must be 1, 2, 3 ,4 ,5 and 6 in some order. So T4, T5 and T6 are 2, 3 and 5 in some order.

c) So E = T4 + T5 + T6 = 10

d) So B + C + D = 45 prisoners - A - E = 24

But B = T4 + 5, C = T4 + T5 + 1, D = T4 + T6 + 4

Substituting gives us 14 = 3*T4 + T5 + T6 = 2*T4 + 10

So T4 = 2

e) T5 and T6 are 3 and 5, in some order

But T6 cannot be 5, because that would make D = 11 = A

So [T1, T2, T3, T4, T5, T6} = {6,1,4,2,5,3}. Final answer

Checking, that makes {A,B,C,D,E} = {11,7,8,9,10}.

Not required, but I notice that no tower contains the same number of guards as a compound contains prisoners.