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Safe and sound (Posted on 2013-07-09) Difficulty: 3 of 5

                +---+                +---+
                | 3 |----------------| 6 |          
                +---+                +---+
               /  |  \              /  |
              /   |   \     D      /   |
             /    |    \          /    |
            /     |     \        /     |
           /      |      \      /      |
      +---+       |       +---+        |
      | 1 |   A   |  B    | 4 |    E  `| 
      +---+       |       +---+        |
           \      |        /   \       |
            \     |       /     \      |
             \    |      /       \     |
              \   |     /    C    \    |
               \  |    /           \   |
                \ |   /             \  |
                 +---+               +---+
                 | 2 |---------------| 5 | 
                 +---+               +---+
The diagram above represents the layout of a new prison.
  1. There are 21 guards, 45 prisoners, six towers for the guards and five compounds for the prisoners.
  2. The towers are numbered from 1 to 6 and the compounds lettered A to E.
  3. All of the compounds are triangular in shape and have a guard tower in each of the three corners.
  4. The number of prisoners in a compound is equal to the total number of guards in the towers on the corners of the compound.
  5. There are 11 prisoners in compound A, one guard in tower number 2, four guards in tower number 3.
  6. At least one guard must be in each tower.
  7. No two towers contain the same number of guards.
  8. 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?

No Solution Yet Submitted by K Sengupta    
Rating: 2.5000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution By the numbers (spoiler) | Comment 1 of 2
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.



  Posted by Steve Herman on 2013-07-09 19:17:37
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