Saturday, October 04, 2008

Math mummies to the rescue

Brian came back the other night with a math question which stumped him, and me. For the first time, I turned to a Singapore mummies' forum where I know of some mums who are superb in mathematics. They came through for me, thanks ladies, if you're reading this.

The question: An emperor wants to marry off his daughter. The way he chooses among the suitors is this. They all sit down at a round table. The emperor walks around the table, saying to the first suitor, "You live". The second suitor is not so lucky, the emperor says "You die", and kills him on the spot with his sword. He lets the third suitor live, but the fourth dies. This goes on around the table till there's only one suitor left.

Figure out where exactly to sit to be the surviving suitor, regardless of how many suitors sit at the table. Use a diagram to help you.


********************************
When I posted the question to the forum, Brian was stuck at this point.

He saw that
a. the pattern was 1,1,3,1,3,5,7,1,3,5,7,9,11,13,15
b. there was a pattern in the interval, ie, [1][1,3][1,3,5,7][1,3,5,7,9,11,13,15]
c. the number 1 is repeated at each binary number
d. each block of interval is also a binary number.

His attempt: If the number of suitors is a binary number, one should go 1st and if it is one less than a binary number, should go last. But he couldn't explain the in-betweens! and that's where he got stuck.
********************************
Mum No1 replied that she and her husband worked out that:

No. of pple 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17........ 32
Best Seat 1 1 3 1 3 5 7 1 3 5 7 9 11 13 15 1 3 ........ 1

We saw that after the # of pple = 3, 7, 15 (before 4, 8, 16 ... all powers of 2) , the cycle will repeat itself in the interval of x 2 ....
Using n as the # of pple, the best seat would be 2(n - the power of 2 which is less than or equal to n) +1
E.g. if n = 9,
Best seat = 2(9-2 to the power of 3) +1
= 2(9-8 )+1
= 2(1) +1
= 3
**********************************
I went WOW, THANKS! Not that I understood lah.

Showed Brian the solution provided by Mum No1. He understood how the pattern worked, ie using the formula, but he still did not really understand the rationale behind subtracting the biggest power of 2. I told him it's cos the intervals start at each power of 2, and we are trying to reduce it so that a pattern can be seen, ie double n and add 1. I was just making a wild stab, cos I too didn't understand the rationale. Of course, he didn't buy my explanation.

He then went on to convert everything into binary system, ie base 2. The pattern seemed more straightforward.

He explained: If we look at the numbers in base 2 (binary), the pattern is seen as, Subtract the biggest binary number (left-most 1) and multiply by 2 and add 1, which in binary system, means moving everything one place left and adding a one at the end (right-most 1):

So,
1 (base 10) = 1 (base 2) --> do above --> 1 (base 2)=1 (base 10)
2 (base 10) = 10 (base 2) --> do above --> 1 (base 2) = 1 (base 10)
3 (base 10) = 11 (base 2) --> do above --> 11 (base 2) = 3 (base 10)
4 (base 10) = 100 (base 2) --> do above --> 1(base 2) = 1 (base 10)
5 (base 10) = 101 (base 2) --> do above --> 11 (base 2) = 3 (base 10)
.
.
.
64 (base 10) = 1000000 (base 2) --> do above --> 1 (base 2) = 1 (base 10)
70(base 10) = 1000110 (base 2)--> do above --> 1101(base 2) = 13(base 10)

Actually, I didn't understand Brian until the next morning, when he had to explain that, "MULTIPLYing by 2 in base 2 is the same as multiplying by 10 in base 10, ie, just put a zero at the back," or move one place value up.
****************************************

So he saw a pattern in base-2, but still couldn't accept the rationale for the solution in base 10.

****************************************
Mum No2 replied:

We figured out the pattern that by subtracting n with the greatest power of 2 less than/equal to n , double the balance + 1.
****************************************

I thanked her and said her solution was very clear and succinct, but Brian still asked "Why subtract power of 2?" aarrggghhhh, I wanted to say, It just is!!
****************************************

Mum No1 replied that she gave the problem to her 8-year old son, who managed to come up with a formula. Amazing kid!

He first listed out the info up to #23 and saw the pattern.
He noticed that when the # of pple is 4, 8 16 .... the best seat would be 1 (the first seat). He then associated them with the power of 2s.
He also noticed that the increment for each cycle is by 2. Eg. [1] [1,3] [1,3,5] [1,3,5,7 ....] This is the part when he thought that there should be a {times 2} in his formula.
Since he didn't know how to "write out" a formula, I told him to describe to me. He started with :-
1) "Find the difference between the no. of pple (n) and the closest power of 2 which is less than n"
2) Then multiply by 2
3) Add 1 (this "1" according to him is to add back the 1st seat)

After making him explained to me like 100x , he wrote down the formula... (He can't wait to go play his lego!)
1+(n-2^) x 2

***************************************
8-year old boy could do it!! Like his mum, I too didn't understand how he came to the solution, but I'm sure he has it all down to pat in his head.

Gahhhh, Brian continued to ask, "I still don't understand why subtract power of 2".
***************************************

After looking at the 8-year old boy's explanation, I kept telling myself, there must be a simple way to explain this. LIGHT BULB appears above my head!!

I explained to Brian, see! see!;

n = 1, 2, 3, 4, 5, 6, 7, 8, ......
nth term = [1][1,3][1,3,5,7][1,3,5,7,9,...][1,....]

The [1,3,5...] sequence would normally have this formula: 2n-1

Now, the sequence restarts when a binary number is reached.
eg for the 3rd interval
n=4,5,6,7
nth term=[1,3,5,7]

What needs to be done is to convert this n so that it starts at 1, and the formula 2n-1 can be used. To do that, we'll need to subtract n by the closest binary number that's smaller than n and add 1.

In this 3rd interval, the binary number is 4, so apply the formula (n-2^)+1

n=4,5,6,7 becomes
n=(n-2^)+1 = 1,2,3,4 (after we subtract n by 4 and add 1)
we can now apply the earlier formula 2n-1 to get the nth term that we wanted.

So what we're doing is.
2n-1
--> 2((n-2^)+1)-1
--> 2(n-2^)+2-1
--> 2(n-2^)+1

Brian smiles and went OHHHHHHHH!!! This is the explanation that he finally really understood fully and accepted.

********************************
I loved how the whole thing panned out. I was getting a splitting headache figuring this out myself, and thanks to some generous mums' help (THANKS LADIES!), Brian now has 2 solutions, one in base-2 and the other base-10.

What I'm most interested now is to see the teacher's solution, cos he's gotta make it simple enough for 5th-graders to comprehend.

20 comments:

Roslyn said...

THANK GOD I'm no longer in school and thank God I have a husband who is good with numbers. If not, my kids sure chum!

Lilian said...

These days it's the adults who are chum...after leaving school for so long, now gotta crack our rusty old brains to help our kids. Already facing difficulty helping them at primary level, dunno what will happen when they get older.

Roslyn said...

Er, private tutors? One of Louisa's classmate has private tuition EVERY day, on top of that, he goes to Kumon.

I am definitely a right-brainer.

Lilian said...

Who's this boy? Private tuition every day? Poor kid.

Do you know tuition didn't cross my mind till you brought it up? I wonder why I hadn't thought of that. Why crack my head when we can pay someone else to do it :)

Tsu Lin + + said...

Gosh, I need some time to digest this. I am very impressed with how the boy figured this out - and yes, somehow they have it right pat in their heads, these people ( I want to say GENIUS!).

My bro is quite impressive with coming up with solutions too. (He's Mensa's top 1%)

Domestic Goddess said...

OMG! I used to think I am very good at Maths till I stumped upon all these difficult Maths scenarios on other mummies' blogs!

Alcovelet said...

Lemme see, I have 10 fingers and 10 toes ... duh, completely lost already. Great job, you mummies! Lilian, you actually understand!

Brian and your 8yo math comrade from over the internet - your abilities are astounding! Good stuff!! (and uh, glad guys you know what you're doing cos I sure don't!)

monlim said...

I asked Lesley-Anne to try it, she couldn't get it. When I was explaining where she went wrong, she got into a huff and just scrolled thru the answer very quickly and said "I don't understand", then stomped off. I think my problem with her is more than just maths!!!

Lilian said...

Tsu Lin: What's your brother's contact? Next time, I'll just enlist his help :) And you're right about the 8-year old boy; these are the type of children I mean when I think Gifted/Genius.

Domestic Goddess: Me too, me too. I always wondered why would anyone have trouble helping their kids in primary mathematics. Till I saw some upper primary Singapore Math. And then I see some of these mummies solving the questions so easily, while I still 'catch no ball' even after looking at the solution. So much for being a Math ace in school!

Alcovelet: If I were reading this off some other person's site, I won't understand either, and wouldn't even bother trying to! But I was trying to find a solution, so put in some effort lah. This was a brilliant question posed by the teacher, and both Brian and I (I more than he) learnt a number of concepts from this exercise that will stay with us forever, I think.

Lilian said...

Haha, Monica, I can just see her stomping off in a huff! Brian's the same if it's something he doesn't understand, he'll just tell me "I don't get it", and stop trying. Grrrr...I think these kids have had it too easy, and never had to think too much to find solutions. You should see Brian when he's asked to do his Chinese tuition work, face like THUNDER, stomp feet to the homework table, and try his luck by slamming door a little too.

Lilian said...

And if it's any consolation Monica, if this weren't homework assigned from school, Brian wouldn't bother trying to understand the solution either, he'd TOTALLY do the same thing as Lesley-Anne, I'm not just saying that to make you feel better.

monlim said...

Phew! I'm glad to hear that. I thot it's just my kid doing the prima donna act :P I don't think it's because she has things too easy, it's an ego thing - she hates to admit she doesn't know something.

Lilian said...

That is one weakness of Brian I'm trying to get him to change: Being afraid to be seen to not know something. His classmates think he has all the answers and it's ego-boosting to him, so my feel is that when he faces difficulties, he'd rather not ask, cos he'll then fall from the pedestal that he's been put on. This fear can be the undoing of many bright kids, as they'd rather not try than to look stupid.

monlim said...

Classic "afraid of being dethroned" first born syndrome!!

Roslyn said...

His mum works full-time so she says she has no time to coach him and his sister. Hence, the private tuition.

Oh, you're smart and clever! Don't need to hire a tutor. Save that money to buy more accessories for your new toy (camera).

Anonymous said...

Hi all,

I had been pondering awhile about this phenomenon of Asian kids consistently beating the rest of the world in global math & science competitions... I'm definitely seeing it here in Vancouver, Canada, when during my daughter's high-school 'Awards Night' the math prizes were INVARIABLY given to Asian boys (mostly from China, HK or in the case of her school, Korea). And in the 5km-radius from where I live we can find about 5 or 6 Kumon schools! (largely filled with Korean kids as my area has the 2nd largest Korean community in Canada, after Toronto).

I was wondering: is it a cultural proclivity, because Math is (at least, up to high-school level..) to a large extent such a skills-&-drill-oriented subject that is therefore so suited to the typical Asian grill-&-drill approach to education..

Then I came upon this website that has many scintillating, provocative (& politically INcorrect, mind you!) articles, one of which seems to shed light on this Asian math & science superiority, claiming that N.E. Asian (i.e. Chinese, Korean, Japanese) have the highest visuospatial IQs (responsible for math & sci abilities) and rank 2nd in overall IQ behind Ashkenazi Jews only because of our lower verbal IQ...

Go hit the website & take a look:
http://www.lagriffedulion.f2s.com/sft2.htm
"SMART FRACTION THEORY II:
WHY ASIANS LAG."

yy. :-)

Anonymous said...

Continuing from the interesting link I posted above (i.e. http://www.lagriffedulion.f2s.com/sft2.htm), I forgot to inform readers that at the bottom of that article one can find a chart correlating ave-national 'verbal IQ' to national GDP, and that on that chart Singapore was ranked HIGHEST in the world for 'verbal IQ'!!

Sort of appears to contradict the premise of that article in that the author was making a point that Ashkenazi Jews have the highest overall IQs and beat the N.E. Asians because of their higher verbal IQs, yet Sgp--which is 75% made-up of pple of NE.Asian descent is ranked highest in verbal IQ.... But perhaps it's because one can't really find a nation whose national-average IQ is a true representation of that of Ashkenazi Jews…?

Anyway, there apparently is, floating around somewhere, evidence to suggest that Sgp-reans have the highest verbal IQs in the world. I know many of us cynical Sgp-reans would immediately jump to the conclusion that aiyah, it's becoz they only picked the smartest kids to take the test, mah! But then, it can really be the true result of a generation of emphasizing English as 1st language in Sgp and the trend of more-&-more English-speaking families in Sgp, even in the minority groups. Of course I’m assuming that the verbal IQ tests were conducted in English because as far as I’m aware, native Mandarin speakers in mainland China or native Cantonese speakers in HK, are very very eloquent talkers & erudite writers in their prospective native tongues too… Not forgetting of course native speakers in East India or any other culture with rich spoken or written literary traditions..

I have some anecdotal examples just glaring at me from within my family about how environment can potentiate genetic predictions: my step-daughter came over to Canada from Sgp in Feb 2007--when half the school-year was already over, and by the end of the school-year she beat the majority-Caucasian school cohort to be Top Student in English!! You see she grew up in a completely English-speaking family and in her preschool years was hot-housed in not fewer than THREE kindergartens one-after-another within the same day (yah, her mom was some champion mommy-coach, sister to that other mom who produced my afore-mentioned-step-nephew-who-could-read-at-age 2). And somehow or other she & her elder brother never had any typical ‘Singlish’ slang rub off on them... Their accent is somewhat ‘neutral’ too, like that of some Sgp newsreaders or LKY or PMLee, for that matter.. And my UK nephew is another example of an Asian kid that prob. beats a lot of british kids in linguistic abilities but then, he did grow up not only in England but also in an entirely English-speaking family--like so many other Sgprean or Peranakan & many Chinese-Malaysian families too... Plus, his dad being a professor in THE Oxford University, he had been going to preschools predominantly attended by kids of other 'academics' & 'intellectuals' too!! His dad was recounting to me one typical language exercise the kids had to do, for instance they had to use different words to describe, say, a sunset. And my nephew came up with: "... That's a vermilion sunset." (what the H*** is a VERMILION sunset for heaven's sake?!? 0_0)

yy.

bACk in GERMANY said...

Wow... can't understand a single word here!!!
This is the reason why I still wake up in cold sweats, whenever my occasional recurring dream of not being able to complete my A level Math paper hits. Thank goodness that is a nightmare...

but no thanks, this is MATH in reality! Yikes!

Gotta go back to work, earn money and pay for a tutor for kids' Math tuition!!!

Lilian said...

haha PP you're funny. My nightmare is always about messing up my exam timetable and being late for exams, or studying for the wrong subject! Hate those nightmares.

bACk in GERMANY said...

Looks like exams have screwed us up for life real badly!