Math Is Fun Forum

  Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °

You are not logged in.

#1 2007-05-02 20:07:20

Identity
Member
Registered: 2007-04-18
Posts: 934

Some questions

Well, I had the test yesterday and I got 4/7 of the questions I think! tongue lol

I can remember two of them I couldn't do, and I was curious to see how they could be done:

6. At a party, 100 people are present, and each of them shakes at least one other person's hand. If someone shakes less than 10 peoples' hands, then they get a red shirt. If someone shakes hands with someone who has a red shirt, they get a blue shirt. Some people are lucky enough to get both a red and blue shirt! Show that Number of Blue shirts < Number of Red shirts.

7. There is a set of natural numbers (i, j, k), where i + j + k = 23. Some examples of sets are (1, 1, 21), (21, 1, 1) and (6, 8, 9). S represents the sum of all possible products i*j*k.
S = 21 + 21 + 432 ...
Show that S is divisible by 23.

These aren't exactly how they were worded, but is approximately how I remember them. Thanks.

Offline

#2 2007-05-02 23:49:28

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Some questions

Well, 7 looks very interesting, but I think 6 doesn't work.

Consider the case when 99 of the people are very socially outgoing and shake any hand that moves, but there's one very shy person who went to the party by mistake because he thought it was a library.

9 of the more charismatic guests manage to convince a handshake out of him, but all the rest of them don't manage it. Meanwhile, they're all quite happy to shake each other's hands.

That means that the number of handshakes made by everyone is as follows:

Shy guy: 9
Charismatic partyers: 99
Everyone else: 98

As he tries to leave, the shy person is presented with a lovely red T-shirt. He mumbles a thankyou and gets out of there as quick as he can. The 9 people who all managed to get a handshake off him are rewarded with 9 lovely blue T-shirts.
(Unfortunately, two of them are drunkenly lost at a Casino later)

So to summarise the events of this party, there were 9 blue T-shirts given out, but only one red.
So there's a rather convoluted counter-example.


Why did the vector cross the road?
It wanted to be normal.

Offline

#3 2007-05-03 09:49:56

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: Some questions

To #7 , I thought of something , don't know it's valid tho.


Numbers are the essence of the Universe

Offline

#4 2007-05-03 16:43:18

Identity
Member
Registered: 2007-04-18
Posts: 934

Re: Some questions

mathsyperson:
Yeah, I had thinking about a similar kind of counter-example when reading through the question, but then I thought maybe I had misread it... in any case I didn't answer it, as I was bothered by the time limit. All the information is definitely present in the question I reproduced, so yeah... I don't know tongue

Stanley_Marsh:
tongue erm... I'm still trying to understand it, looks interesting

Offline

#5 2007-05-04 02:49:18

Stanley_Marsh
Member
Registered: 2006-12-13
Posts: 345

Re: Some questions

OH , I don't know , maybe it's not true that

I just thought , i,j,k's range are equal , (i,j,k)=(1,1,21),(1,2,20) , by switching the place (i,j,k)=(21,1,1) or (1,21,1) , so I think , i can have one 21 , j can , k can also , so their sums are equivalent?

Last edited by Stanley_Marsh (2007-05-04 02:49:33)


Numbers are the essence of the Universe

Offline

Board footer

Powered by FluxBB