Monday, September 29, 2008

Aug-Sep Mystery MathAnagram Revealed - The Incomparable Poincare!

If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living. Poincare

The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. Poincare

Finally! Jules Henri Poincare!
Considering the stature of this brilliant individual and his contributions to mathematics, physics and philosophy, it's only fitting that he occupy our contest for at least two months (and i will leave his picture in the sidebar for awhile.). The standard reference for biographies will give you detailed background. Look here. For more of Poincare's profound quotes look here.

Our Winners Are:


Sean's Contribution:

(1) Henri Poincare
(2) He is described as a "polymath" and has been called "The Last Universalist" because he excelled at all the established mathematical fields of the time.
Also, Prince Louis-Victor de Broglie was awarded the Poincare Medal ( and the Prince of Monaco attended Poincares funeral (

Steve's Contribution:
1. Henri Poincare
2. Grigori Perelman has been credited with proving the Poincare Conjecture, although not without controversy. Perelman refused the Fields Medal offered to him after his proof was confirmed.
3. The clues that I used were in the Aug. 26 post, the word "relatively" and the reference to the "last of a dying breed." Poincare worked on relativity and is considered the last universalist. I also think the phrase "truly unique" refers to Poincare being a jewel; his first name is Jules.

If you're thinking that I feel reverence for Monsieur Poincare, you would be correct...


Anonymous said...

Hi Sir,

I have a Maths bloopers that needs your assistance. This question was set in a renowned primary school in Singapore. This question was a killer....Really appreciate if you can solve the sums without using algebra.

Reflected depicts the question:

There are some marbles in Box A and Box B. If 50 marbles from Box A and 25 from Box B are removed each time, there will be 600 marbles left in Box A when all marbles are removed from Box B. If 25 marbles from Box A and 50 marbles from Box B are removed each time, there will be 1800 marbles left in Box A when all marbles are removed from Box B. How many marbles are there in each box?

The question below is also challeaging but students are not allowed to use algebra to solve this. They are taught using model diagram instead.

Andrea has $200 more than Bala. Andrea gives 60% of his money to Bala. Bala then gives 25% of his money to Andrea. IN the end, Bala has $200 more than Andrea. How much did Andrea have at first?

Alex said...

Here's a solution to the first problem using words, though it's basically equivalent to Algebra:

Let's suppose you knew how many marbles were in Box B. Then the difference between 1800 and 600 is the same as the difference between twice that number, and half that number. So you need a number whose half and whose double are 1200 apart. Guesswork gives you the answer as 800.

Alex said...

For problem 2, I'll do it in pictures, though I've never tried that method before.

A has ---------- & $200
B has ----------

After A has given away 60%:
A has ---- & $80
B has ---------------- & $120

After B has given 25% back:
A has -------- & $110
B has ------------ & $90.

Now, at the end we're told that Bala has $200 more. So, those four blocks extra must be worth $220... meaning one block is worth $55, so Andrea started with $750.

Hope that helped. It's just a case of drawing the situation in the start, and then seeing how it progresses. As a disclaimer, whilst I didn't use Algebra to solve either of these, I do have Algebraic training, which gives me a major advantage over those primary school students.

Dave Marain said...

Anon and Alex--
I guess it's appropriate to discuss problem-solving in tribute to Henri Poincare! These two problems are so nice they should probably live on their own in a separate post. I may do that.

Alex, I was thinking along your lines but I was trying to imagine the kind of 'model diagram' used by Singapore students. It's difficult for me not to use algebraic methods, although the model diagrams seem to be based on the concept of a common unit in ratio problems which is equivalent to a variable IMO.

Anyway, here is my crude attempt for the marbles problem. Let me know if it makes sense:

First, without modeling, I concluded from Box B being emptied each time, that the NUMBER of removals the first time was TWICE the NUMBER of removals the second time. This could probably be demonstrated visually but I didn't.

From this I concluded that, the first time, Box A had TWICE as many removals with TWICE as many marbles removed each time. In other words, the first time Box A had FOUR times as many marbles removed as the second time. Note that this ratio depended on what was removed rather than what was left. I think that's critical.
Now for model diagrams:
These will refer to Box A only. The top model is for the first removal.
Therefore, each unit, |----|, represents 1200 divided by 3 or 400 marbles. From this, I concluded that the total number of marbles for Box A is 1800 + 400 = 2200. I'll omit the calculation for Box B.
I'm not at all sure these models make sense or if my reasoning is clear - but it's an attempt. I'd be interested in hearing from those far more experienced with Singapore models than I am.