Last Monday, the Maths Department, or maybe just Mr Alvin Lim himself, posed some Maths questions and we have to send our solutions to the school. The deadline is today at 12 noon. They claimed that prizes will be given to the best solutions from each level.

It was Wei Liang who made this announcement on stage, because he is the Maths Society chairman. He was very smart to make an announcement about Chinese Orchestra concert as well, which he told me later that it was quite impromptu.

Anyway, the 2 questions are:

1) The Lost World
In a Mathematics class,
70% of the pupils have lost their pen,
75% of them have lost their books,
80% of them have lost their calculators, and
85% of them have lost their rulers.

At least what % of the pupils have lost all 4 items?
[Picture of the Jurassic Park - The Lost World]


2) Bond Concert
The Bond Girls' concert starts in 16 minutes tonight. They need to cross an old bridge and can only have 1 torch with them. Only 2 Bond Girls can cross at one time and each time, one has to return with the torch to bring another acorss.
The time taken for each girl is as follows:

Haylie - 1 minute
Eos - 2 minutes
Tania - 4 minutes
Gay-Yee - 8 minutes

The time taken for 2 girls to cross the bridge is that of the slower one. Can all of them cross in 15 minutes?

I have the solutions myself already, but I am not posting them to the school. I am posting them here instead haha.

So for 1st question,

We have a certain percentage of students losing all 4 items, and we want to keep it to the very minimum. In order to do this, we must have as many people losing only 3 out of 4 items, while the rest losing all 4 items. So something like

+--------+--------------------+
| Lose 4 |    Lose 3 items    |
+--------+--------------------+

(Note that if it is possible for all students to lose 3 items, or if it is possible for some students to lose 3 items and the rest to lose 2 or less items, then the answer to the question is 0%)

Now to find as many people as possible to lose all 3 items is no mean feat. Simply change the statements to:
30% of the pupils did not lose their pen (item A),
25% of them did not lose their books (item B),
20% of them did not lose their calculators (item C), and
15% of them did not lose their rulers (item D).

So now we have to make sure that we have as many students NOT losing only 1 item, as possible. We can represent it by a model:

+----------+-------+--------+----------+-----------+
| Lost all | DNL D | DNL C  |  DNL B   |   DNL A   |
+----------+-------+--------+----------+-----------+
              15%      20%       25%        30%
*DNL = Did not lose

From there it is obvious that those students under "DNL D, DNL C, DNL B and DNL A" lose 3 out of the 4 items. And number of students who lost all 4 items is kept to a minimum (no matter how hard you try you always get the same number of students). Hence the percentage of students who lost all 4 items is 100% - 15% - 20% - 25% - 30% = 10%.

Many people got 45% for their answer because they used only the lowest 2, ie: 70% and 75%, and by using the same method, they found the answer of 45%. However, 80% and 85% of those who lost C and D respectively can also be part of those who did not lose all items, and hence those who lost all 4 items can be minimised further.

Well as for the 2nd question, through some thinking 2 solutions can be found:

4,8 -------- 1,2
4,8,1 ------ 2
1 ---------- 2,4,8
1,2 -------- 4,8
----------- 1,2,4,8

The 2nd solution is simply swopping Haylie and Eos (1 and 2 minutes respectively), which it will end up with 15 minutes too.

The solution lies with Haylie and Eos being the "transporters" of the torch. Being the fastest 2 people, both of them are the ones to bring the torch back and forth. Of course, the first step is to bring both of them across so that there is someone at the other end to return the torch after Tania and Gay-Yee (4 and 8 minutes respectively) have crossed.

There is one more question for staff too, which Mr Lim said that there IS a pattern.

A, B, H, F, M, C, I, G, T, D, O, J, U, E, X, _?

I still have not solved this question so maybe going to figure out. Tried converting all to numbers, which did not show any pattern at all. Noticed that all the letters are being used only once, and the remaining letters are: K, L, N, P, Q, R, S, V, W, Y, Z. One lame solution brought up was that the answer is A, because the pattern repeats itself after the X.