jcmc2112

Let's start with this one:

98 points are given on a circle. Alice and Billy take turns drawing a
line segment between two of the points which have not yet been joined by a
segment. The game ends when each point has been used as the endpoint of a
segment at least once. The winner is the player who draws the last segment.
If Alice goes first, who has a winning strategy? Why?

iamsounsure

Alice and Billy should find something more fun and productive to be doing.

Problem solved.

jcmc2112

iamsounsure should find something even more productive than trying to be funny and failing at it.

First person who gets the correct answer gets a virtual handshake by me.

math+sci=asian

Ok, I'll take a crack at it:

So we have a 98-gon, so number of diagonals would equal: 98(95)/2=4655

4655-1=4644, therefore Alice has a winning strategy because it's an even number. Easy problem.

jcmc2112

Correct!

Next:

Each of ten boxes contains a dierent number of pencils and there is at least
one pencil in each box. No two pencils in the same box are of the same color.
Prove that one can choose a collection of ten pencils, one from each box, so
that no two of the ten pencils in the collcetion are of the same color.

LogicWarrior

um

If the numbers of pencils in each box are different there has to be at least 1 in the first, 2 in the second, 3 in the third, etc. Not those numbers exactly but box 1<box 2<box 3 etc.

Since no two pencils in each box are the same then if the first box has just red, the second box must have a different color since it has more pencils, box 3 must have an additional new color since it has more pencils, and so on.

As long as you get to look in the boxes, you can get a different colored pencil from each box.

best way to spend a friday night people, join in

jcmc2112

You explained what is being asked but it still isn't answered

LogicWarrior

Yes it is. Read it again.

iamsounsure

He explained it perfectly. Just in terms of words not mathematics.

trout

order the boxes by number of pencils, then choose a pencil from each in order. At the Nth box you will have N-1 pencils in hand, with greater than or equal to N pencils of unique color in the box so there must be at least one new color to choose.