Math Help - General Problem Solving

1. 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.

2. 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