# General Problem Solving

• Feb 9th 2009, 11:06 PM
BG5965
General Problem Solving
Hi, I have 2 problems which I would like help with.

1) A captain commands 4 ships which can hold 3 tonnes of cargo each. His client has a number of boxes:
i] None are heavier than 1 tonne
ii] Together, they weigh 10 tonnes.

Show that it may not be possible for him to carry away all the cargo at once with four ships, but with 5 ships, it is always possible. (So basically find a combination of boxes, that follow the rules above, that four ships cannot carry at once).

2) 8 schools are part of a competition. Each school versus every other school exactly once. Prove that at any time, there are at least two teams that have versed the same number of schools.

• Feb 10th 2009, 09:06 PM
CaptainBlack
Quote:

Originally Posted by BG5965
Hi, I have 2 problems which I would like help with.

1) A captain commands 4 ships which can hold 3 tonnes of cargo each. His client has a number of boxes:
i] None are heavier than 1 tonne
ii] Together, they weigh 10 tonnes.

Show that it may not be possible for him to carry away all the cargo at once with four ships, but with 5 ships, it is always possible. (So basically find a combination of boxes, that follow the rules above, that four ships cannot carry at once).

If they all weigh the same then for some integer $N>10$ they each weigh $10/N$ as $N$ of them must weigh $10$ tonnes and they each weight $\le 1$ tonne.

Each ship can carry $K=\lfloor 3/(10/N) \rfloor$, and

$4 \times K \times (10/N)<10$

while:

$5 \times K \times (10/N) \ge 10$

Now use guess and check to find a suitable $N$.

CB