This puzzle also comes courtesy my eldest nephew.

There are 101 boxes. A girl marked them “1”, “2”, “3”…. all the way to “101”. She had two colored pens to mark the boxes with – red and blue. And she used both of them. Only one color was used for any one box. We know that the total number of blue boxes is the same as the highest numbered blue color marked box. And to that if you add twice the lowest numbered red color marked box, you will get the total number of boxes.

The question is, how many blue and red boxes were there?

How many countries are completely surrounded by water? The count completely surprised me.

Can you name most of them?

In American football, any drive can end with the following points – 0 (ball turned over), 2 (safety), 3 (field goal), 6 (touchdown with failed point after or failed two pointer), 7 (touchdown and successful point after) or 8 (touchdown and successful 2 point conversion).

Assuming no team scores more than 50 points in a playoff match (there has to be a winner), what are the different legitimate combinations of scores that a match can possibly end with? (e.g. a game can end 14-2 but can never have a 27-1 score).

The easiest way to think about this problem is not to think of number of marbles in each color but the differences in number between colors. Let’s say “a”, “b” and “c” are the pairwise differences in colors. Their values are 10, 10 and 20 to begin with.

Let’s see what happens whenever there is a “hit”.
If the marbles are of same colors, of course no change in count happens – so a, b and c remain the same.
If the marbles are of different colors – then those two colors go down by 1 each and the third color goes up by 2. Which means the pairwise differences either remain the same (the two colors that hit) or they change by 3 (one color goes down by 1 and the resulting color goes up by 2).

So, the first concept to realize is that a, b and c always change in steps of 0 or steps of 3. There are no other step changes they can have.

Now, let’s say what if the marbles actually did become all of one color. What as the step right before it? We would have had to have all marbles of the same final color excepting two – one each of the other two colors. Thus if finally everything is green, in the penultimate step, there has to be one red, one blue and rest green. The red hits the blue to become green.

Note that in that penultimate step, the pairwise difference between two colors (that have one marble left each) is 0. And that is the second concept to realize.

Now putting these two concepts together…

Well, we started with a, b and c being 10, 10 and 20. Going in steps of 0 and 3, can any of these numbers ever get to 0? No. Therefore it will never reach the penultimate step.

Which means the marbles will never be of the same color !!

There is a large tray with red, blue and green marbles in it. In fact, there are 20 red marbles, 30 blue marbles and 40 green marbles. In that tray, the marbles are continuously moving around in Brownian (random) motion – often hitting each other.

If two marbles of same color hit each other, they just bounce off and continue. However, if two marbles of different color hit each other, then, both change to the third color and then bounce off and continue. All such hits are always between two marbles only. (no simultaneous three or more marbles hitting each other).

So a red and blue marble hitting each other will make both of them turn green and so on….

Is it ever possible for all the marbles in the tray to become of one color? If so, which color will it be?

This math puzzle about Covid will make your jaw drop (also make you feel better after getting a test)

First, please take all precautions again Covid. If you think you got exposed or have symptoms, get yourselves tested, quarantined and everything else CDC is asking us to do. Once you have done so, think about this puzzle…

The following puzzle – which is a very real one in Covid days – is an adaptation from a puzzle I had posted last year.
The end question I am going to ask you is this:
Q1. Your Covid test came positive. What is the probability that you really have the infection?
Q2. Your Covid test came negative. What is the probability that you really do not have the infection?

As Narayan had pointed out last year, most doctors do not get this. The fact that all tests have false positives and false negatives (they are not perfect) makes the real probability unintuitively high or low.

The real answers will knock your socks off.

Ok, let’s start this:
——
1. The COVID PCR test has a false positive of about 0.5% (*See Source). Which would mean in about 200 people without the virus, the test will make a mistake in one case and say he/she has the virus.

2. The test has a false negative of 10% (** See Source) This means of 10 people who actually have the virus, the test will miss out 1 of them.

3. Assume 2% of Americans at any point of time has the virus (*** See Source)

You went to get a PCR test done. Test showed positive. What is the real probability you have the virus? Hint: It is NOT 99.5%
Take the other case – the test showed negative. What is the real probability you do not have the virus? Hint: It is NOT 90%

Both the answers will make you feel much better.

That should be good enough reason why we should LOVE math! And at the risk of Tammi – the math teacher in my friend circle – slapping my wrist, I would say, you will love the chapter on Conditional Probability.
——

Sources:
* (in reality 0.2%-0.9% per https://www.icd10monitor.com/false-positives-in-pcr-tests…)
** the actual number can widely vary – as high as 67% – depending on when you got tested after exposure. The more time you give after exposure, the more the virus gets to establish itself and lesser the false negative number. https://www.acc.org/…/variation-in-false-negative-rate…
*** I took CDC’s 25% estimated US population had it https://www.cdc.gov/…/2019-ncov/cases-updates/burden.html over 11 months and took the average as 2%. (It will be slightly higher due to ramp ups)