Brain Teaser

You have 12 pearls. 11 of them are real, and 1 of them is fake. They all look/feel/everything EXACTLY the same, the only difference between the real and fake ones is their density.

You are given a balance scale (one of those old school scales that compare weights. You can weigh/compare the pearls three (3) times. Describe how you would:

1) determine the fake pearl

2) determine whether the fake pearl is more or less dense than the real pearls.

Post your answer in the comments!

3D drawing software - Google Sketchup

Found this rather interesting (and free) software: Google Sketchup. Interesting because it enables you to draw in 3 dimension as well as manipulate it around to give you views from different angle.

Perfect for those who have trouble visualising 3D trigonometry problems.

A 'simple' maths problem

A teddy bear has a cost price of $18, and a selling price of $21. One day, a young man decided to buy that bear and gave the shopkeeper a $100 dollar bill.

Unfortunately, the shopkeeper do not have $100 change. Hence he went to change the $100 bill into smaller change from a passer-by and gave the change to the young man.

After some time, the passer-by realise the $100 is a fake and demanded his money back from the shopkeeper! The shopkeeper has no choice but to return $100 to the passer-by.

Question: how much exactly did the shopkeeper lost?

Probability in DotA?!?!

Yes, you may have known that Mr Guay is currently a little into the game of DotA. And here are some interesting items in DotA that poses some mathematical questions!

There is an item in the game called Vanguard. Here are the details, taken from DotaPortal:

Vanguard

Description:+5 HP regen/second
+250 Hit Points
65% to block 40 damage

What's interesting is that the amount of damage it blocks is actually a probability! Now...what if you have 2 Vanguards? What is the effect? Some choices:

a) 65% chance to block 80 damage
b) No additional effect i.e. same as having one Vanguard. Which means you just wasted valuable gold!
c) 65%+65%=125% (or rather 100%, since probability can not exceed 100%) to block 40 damage. This means that you get a permanent block of 40 damage.
d) The 2 Vanguards are mutally independent, and hence we need to recalculate the probability.
e) I don't care cos games and maths don't mix! Stop spoiling it!

use your mouse and highlight the space below for answer....

Although Mr Guay can't be 100% sure as he is not the games programmer, it will most likely be (d). So to calculate the probabilty, we can do the direct method, i.e.

P(both Vanguard activating) + P(1 Vanguard activating and the other not)
= (0.65 x 0.65)+[(0.65x0.35)+(0.35x0.65)]
= 0.8775
= 87.75%

Or indirectly,
1 - P(both Vanguards not activating)
= 1 - (0.35x0.35)
= 0.8775
= 87.75%

So, when we buy 2 Vanguards, we increase the probability from 65% to 87.75% of blocking 40 damage. Pls note that Mr Guay is not sure how much damage 2 Vanguards will block when both activate but my guess is only a block of 40 damage instead of 80 damage.

That's all well and good but in the game, the amt of gold is quite limited. There is another item which is one of the material used to make a Vanguard and has similiar properties. Let's compare them and their price:

Vanguard ($2275) 65% to block 40 damage

Stout Shield ($300) 30% to block 30 damage

So which is more worthwhile? How many Stout Shields do we need to buy to obtain the same probability of damage reduction of 1 Vanguard? I know the amt of damage reduction is lesser but disregard that for now. Also disregard the additional porperties Vanguard gives you such as HP regeneration, as we're only concern about blocking damages.

use your mouse and highlight the space below for answer....

Probability of shield NOT activating = 1 - 0.3 = 0.7 (or 70%)

P(2 stout shields)
= 1 - P(both shields do not activate)
= 1 - (0.7 x 0.7)
= 0.51
= 51%

P(3 stout shields)
= 1 - P(3 shields all do not activate)
= 1 - P(0.7 x 0.7 x 0.7)
= 0.657
=65.7%

Hence, with 3 Stout Shields we can have the same blocking probability as a Vanguard, but 3 Shields only cost $900! Something to think about....

There are some other items in DotA that deals with probability, such as Butterfly, Basher, Buriza etc. Have fun!

Games to help in coordinate geometry!

Found this 2 great games that help with coordinate geometry!

Game 1 - Catch the Fly

For students who are still very confuse which is x, and which is y in coordinates? (2,3) or (3,2) leh?

Link: http://hotmath.com/hotmath_help/games/ctf/ctf_hotmath.swf

Game 2 - Equation of line aka Kill the Cockroaches!

This is a great game! Form equations of the stright line correctly by looking at the path of the cockroaches and send them to hell!

Link: http://hotmath.com/hotmath_help/games/kp/kp_hotmath_sound.swf

Other great games here: http://hotmath.com/games.html

Very Interesting Answers Pt 3 - Graphical Humor

I think I deserve some comic relief after marking the mid-yr papers. And these sure put a smile onto my weary face! I'll show the correct graph, followed by the hilarious mistakes.

Type 1 -Quadratic curve.

This is the correct answer:

This student manage to draw a straight line....don't ask me how!

Type 2 - Histograms

Correct answer:

Variant 1: artistic impression of a histogram

Variant 2: I don't even know how the student plot this graph!

Type 3 - Cumulative Frequency Curve

Correct graph looks like this:
Variant 1: wrong axis produces interesting curves

Variant 2: "I love curves, let's do more!"

Variant 3: I don't like curves:

Last but not least, though not related to graphs, it's my personal favorite:

Very interesting answers Pt 2

First, let's start with some 'reruns' of mistakes....

Formulae get mixed up because they memorise the letters but not the definition!
Potential difference and Power both begin with 'p', hence...

Some have the audacity to repeat the exact same mistake as last time!

I'm not sure why but my students REFUSE to read the question. Even after I bold the words 'label all components'. Less than 10 out of 46 students actually labelled the diagram.

Another example of not reading the question. But I guess that's less condescending than assuming they do not know the meaning of 'vertical' and 'horizontal'?

Some are clearly confused... almost to the point of nonsensical.

I'm not sure to burst out in laughter or curl up and cry when I saw this...

"This way,I can at least get 1 of it correct!"
Yah, but it also means you will definitely get the rest of it wrong!


Not that I condone copying, but could they at least put some effort in disguising the fact they've copied?

my ears feel wet...is it blood oozing out?