JOSEPH'S BLOG: THE FAST AND THE FURIOUS: TOKYO DRIFT

Monday, November 13, 2006

2
4
8
4000(Ans of the fifth qn)
FIFTH QN:4000(Ans of the tenth qn)
2000
SEVENTH QN:4000
6000
4000(Ans of the fifth qn)
TENTH QN:4000(Ans to the seventh qn)

Cristo puede mover montes Solo Dios puede salvar mi Dios puede salvar por siempre autor de salvación Jesus la muerte vencio El la muerte vencio
3:19 AM

What is 1+1?
What is 2+2?
What is 4+4?
What is the answer to the fifth question?
What is the answer to the tenth question?
What is 1000+1000?
What is 2000+2000?
What is 3000+3000?
What is the answer to the fifth question?
What is the answer to the seventh question?

Cristo puede mover montes Solo Dios puede salvar mi Dios puede salvar por siempre autor de salvación Jesus la muerte vencio El la muerte vencio
3:14 AM

There are 12 posibilities.

There are 3 boxes, and we can change the position of the piles(so that it is 3! which is 6), so that means there are actually 2 different 'ways' of having the piles.

The WHY-DIDNT-I-THINK-OF-IT Method:
1 box has (1), another box has (100), and another box has the cards 2-99

If the sum is 1-100, it means it is from the 1st and 3rd box
If the sum is 101, it means it is from the 1st and 2nd box
If the sum is 102-199, it means it is from the 2nd and 3rd box

Why didn't you think of it?

The I-SERIOUSLY-CANNOT-THINK-ABOUT-IT Method:
Place 1, 2, 3 in different boxes (6 possibilities) and then place n in the same box as its residue mod 3

Don't understand? This is SUPPOSED to help you understand...trust me...it won't

Let Hn be the corresponding result that for cards numbered 1 to n the only solutions are by residue mod 3, or 1 and n in separate boxes and 2 to n - 1 in the third box. It is easy to check that they are solutions. Hn is the assertion that there are no others. H3 is obviously true (although the two cases coincide). We now use induction on n. So suppose that the result is true for n and consider the case n + 1.
Suppose n + 1 is alone in its box. If 1 is not also alone, then let N be the sum of the largest cards in each of the boxes not containing n + 1. Since n + 2 ≤ N ≤ n + (n - 1) = 2n - 1, we can achieve the same sum N as from a different pair of boxes as (n + 1) + (N - n - 1). Contradiction. So 1 must be alone and we have one of the solutions envisaged in Hn+1.
If n + 1 is not alone, then if we remove it, we must have a solution for n. But that solution cannot be the n, 1, 2 to n - 1 solution. For we can easily check that none of the three boxes will then accomodate n + 1. So it must be the mod 3 solution. We can easily check that in this case n + 1 must go in the box with matching residue, which makes the (n + 1) solution the other solution envisaged by Hn+1. That completes the induction.
My much more plodding solution (which I was quite pleased with until I saw the more elegant solution above) follows. It took about half-an-hour and shows the kind of kludge one is likely to come up with under time pressure in an exam!
With a suitable labeling of the boxes as A, B, C, there are 4 cases to consider:
Case 1: A contains 1; B contains 2; C contains 3 Case 2: A contains 1,2 Case 3: A contains 1, 3; B contains 2 Case 4: A contains 1; B contains 2, 3.
We show that Cases 1 and 4 each yield just one possible arrangement and Cases 2 and 3 none.
In Case 1, it is an easy induction that n must be placed in the same box as its residue (in other words numbers with residue 1 mod 3 go into A, numbers with residue 2 go into B, and numbers with residue 0 go into C). For (n + 1) + (n - 2) = n + (n - 1). Hence n + 1 must go in the same box as n - 2 (if they were in different boxes, then we would have two pairs from different pairs of boxes with the same sum). It is also clear that this is a possible arrangement. Given the sum of two numbers from different boxes, take its residue mod 3. A residue of 0 indicates that the third (unused) box was C, a residue of 1 indicates that the third box was A, and a residue of 2 indicates that the third box was B. Note that this unique arrangement gives 6 ways for the question, because there are 6 ways of arranging 1, 2 and 3 in the given boxes.
In Case 2, let n be the smallest number not in box A. Suppose it is in box B. Let m be the smallest number in the third box, C. m - 1 cannot be in C. If it is in A, then m + (n - 1) = (m - 1) + n. Contradiction (m is in C, n - 1 is in A, so that pair identifies B as the third box, but m - 1 is in A and n is in B, identifying C). So m - 1 must be in B. But (m - 1) + 2 = m + 1. Contradiction. So Case 2 is not possible.
In Case 3, let n be the smallest number in box C, so n - 1 must be in A or B. If n - 1 is in A, then (n - 1) + 2 = n + 2. Contradiction (a sum of numbers in A and B equals a sum from C and A). If n - 1 is in B, then (n - 1) + 3 = n + 2. Contradiction ( a sum from B and A equals a sum from C and B). So Case 3 is not possible.
In Case 4, let n be the smallest number in box C. n - 1 cannot be in A, or (n - 1) + 2 = 3 + n (pair from A, B with same sum as pair from B, C), so n - 1 must be in B. Now n + 1 cannot be in A (or (n + 1) + 2 = 3 + n), or in B or C (or 1 + (n + 1) = 2 + n). So n + 1 cannot exist and hence n = 100. It is now an easy induction that all of 4, 5, ... 98 must be in B. For given that m is in B, if m + 1 were in A, we would have 100 + m = 99 + (m + 1). But this arrangement (1 in A, 2 - 99 in B, 100 in C) is certainly possible: sums 3 - 100 identify C as the third box, sum 101 identifies B as the third box, and sums 102-199 identify A as the third box. Finally, as in Case 1, this unique arrangement corresponds to 6 ways of arranging the cards in the given boxes.

Cristo puede mover montes Solo Dios puede salvar mi Dios puede salvar por siempre autor de salvación Jesus la muerte vencio El la muerte vencio
3:03 AM

Sunday, November 12, 2006

btw... imo exams are super hard...

u've got 4h 30 min to answer 6 qn

and ea qn contains 7 marks

i chose the easier ones so u ppl would actually understand

A magician has 100 cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card.
A member of the audience selects two of the three boxes, chooses one card from each and announces the sum of the numbers on the chosen cards. Given this sum, the magician identifies the box from which no card has been chosen.
How many ways are there to put all the cards into the boxes so that this trick always works?(Two ways are considered different if at least one card is put into a different box.)

Cristo puede mover montes Solo Dios puede salvar mi Dios puede salvar por siempre autor de salvación Jesus la muerte vencio El la muerte vencio
10:26 PM

betcha can't answer this qn...

got it from www.nzmaths.co.nz

dun cheat and get the answer frm there

Richard finds 9 dice arranged in a 3 by 3 square. He’d like to rearrange the dice so that only 6's are showing. But there are rules as to how he can move the dice. He is only allowed to pick up a row of three dice or a column of three dice and rotate them altogether in the same way. Is it possible for Fred to accomplish this task? Can you tell him how?

Cristo puede mover montes Solo Dios puede salvar mi Dios puede salvar por siempre autor de salvación Jesus la muerte vencio El la muerte vencio
9:52 PM

I give you a clue... a=b!!!
Remember that...

Cristo puede mover montes Solo Dios puede salvar mi Dios puede salvar por siempre autor de salvación Jesus la muerte vencio El la muerte vencio
9:34 PM

QN1) There was a bag of marbles. A took 1 marble and 1/5 of the remaining marbles. B,C,D,E and F did the same. What is the second least possible number of marbles there were in the bag at first?

Cristo puede mover montes Solo Dios puede salvar mi Dios puede salvar por siempre autor de salvación Jesus la muerte vencio El la muerte vencio
9:32 PM

THE ZERO PROOF

a=b
a*a = a*b
a*a - b*b = ab - b*b
(a-b)*(a+b) = b*(a-b)
(a+b) = b
a+b =b
a+b-b=b-b
a=0

* = multiply

Cristo puede mover montes Solo Dios puede salvar mi Dios puede salvar por siempre autor de salvación Jesus la muerte vencio El la muerte vencio
3:04 AM

4 MATHS GENIUSES ONLI!!!KEEP OUT IF UR NOT 1...U'LL GO CRAZY

Cristo puede mover montes Solo Dios puede salvar mi Dios puede salvar por siempre autor de salvación Jesus la muerte vencio El la muerte vencio
3:03 AM

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My awards(tks xy)

SMOPS-2005-GOLD
2006-PLATINUM
1st in maths 2004 and 2005

AWARD FOR BEST COMMENTER

XIAO YING[1]

Bible Verse

1 Timothy 6:18
Command them to do good, to be rich in good deeds, and to be generous and willing to share.

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Archives
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Designer
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Band ppl

Yi Rui:teaching him to be better than nicholas

Douglas:Watching him attempt to grab cass' arm.

Nicholas:Making him do stars and using de cert to double it

Cass:Making her carry stand, box...etc. and saying it's on her honour

Denise:Super l8 nite calls, calling her mrs teo,xiao xian...etc.

Lianne:Scolding her 4 wake-up sms/calls, and scolding her 4 trying to make us deaf with her super-deafening voice, calling her xiao ying[2], talking to her bro in chn...lolz

Ezbon: de BLURNESS in band camp, saying ty sir twice when it's not his turn to go

Ming Jian:juz chatting abt stuff.

Bryan:oso chatting abt stuff.

Instructors: their teachings, and sometimes scoldings

If ur name is not here, cuz i dunno so much abt u...but i'll miss all de band members

Classmates

Jodi: kicking me and whacking me and pinching me and poking me? no i won't miss her...hahaz

Gloria: calling her mrs goh, especially in de ge performance

Tessa: calling her xiao qi

Ying Xin: calling her xiao ying

Johnny:chatting abt maple stuff, calling him xiao xian

Kai Jun:ahhh...de maple freak, calling him xiao kai and messing up with his hair

Joshua: some v weird jokes, teasing him tt he got bronze 4 smops, sharing secrets which he told every1(i wun miss him 4 tt)
Caleb(i noe he's not my class but heck):sharing mathsy stuff

Joel: getting him into trouble by reporting him to tchrs...hahaz

Waggly: sharing some secrets, chatting, going to his house 4 pt...lolz, and calling him wag, seeing him wag his legs while singing de sch song...HAHA

Shao Yun:err...dunno wad to say...mayb juz chatting?

Pheng Hwee: i dunno, toking abt maple, which he hardly plays, and i don't play anymore

Azzac: scolding him in chn, which turns out to be scolding myself, cuz we hav de same chn name...hahaz

Wei Jie:chatting...discussing abt lots of stuff, cuz we are like mostly together in projects.

Bryan(LEE):i dunno...debating with him wad's de meaning of 'with'

Bryan(TAN):watching him play fantasi impromptu

Glendon:my human phone book...miss getting all de no. frm him

Kester: chatting alittle i guess...

Tchrs: their tching which can sometimes lead to scolding which sumtimes are really weird tt i laugh and i get scolded 4 laughing

If ur name is not here, cuz i dunno so much abt u...but i'll miss all de classmates

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