Tag Archives: fun with Maths

How to Remember a Round Number

A traditional French rhyme goes like this*:

Que j’aime a fatre apprendre

Un nombre utile aux sages!

Glorieux Archimede, artiste ingenieux,

Toi, de qui Syracuse loue encore le merite!

( * A loose translation is:

How I like to make

The sages learn a useful number!

Glorious Archimedes, ingenious artist,

You whose merit Syracuse still praises.)

But to which ‘number useful to the sages’ does it refer? Counting the letters in each word, treating ‘j’ as a word with one letter and placing a decimal point after the first digit, we get

3.141 592 653 589 793 238 4626

which is \pi to the first 22 decimal places. Many similar mnemonics for \pi exist in many languages. In English, one of the best known is

How I want a drink, alcoholic, of course, after the heavy 

chapters involvlng quantum mechanics. One is, yes,

adequate even enough to induce some fun and pleasure

for an instant, miserably brief.

It probably stopped there because the next digit is a 0, and it’s not entirely clear how best to represent a word with no letters. Another is

Sir, I bear a rhyme excelling

In mystic force, and magic spelling

Celestial sprites elucidate

All my own strivings can’t relate.

An ambitious \pi -mnemonic featured in “The Mathematical Intelligencer” in 1986 (volume 8, page 56). This is an informal “house journal” for professional mathematicians. The mnemonic is a self-referential story encoding the first 402 decimals of \pi. It uses punctuation marks (ignoring full stops) to represent the digit zero, and words with more than 9 letters represent two consecutive digits — for instance, a word with 13 letters represents the digits 13 in that order. Oh, and any actual digit represents itself. The story begins like this:

For a time I stood pondering on circle sizes. The large 

computer mainframe quietly processed  all of its assembly 

code Inside my entire hope lay for figuring out an elusive

expansion. Value pi. Decimals expected soon. I nervously

entered a format procedure. The mainframe processed the

request. Error. I again entering it, carefully retyped. This

iteration gave zero error printouts in all — success.

You can find out more about \pi related mnemonics in various languages please use the internet.

More later,

Nalin Pithwa

Coffee time mathematics — any number via three twos

Problem:

Here’s a witty algebraic brain teaser that had amused participants of a congress of physicists in the erstwhile USSR. The problem is to represent any number that must be positive and whole (any positive integer) using three twos and mathematical symbols.

Solution:

Let us take a particular case, and think “inductively”. Suppose we are given the number 3.  Then, the problem is solved thus:

3=-\log_{2} \log_{2} \sqrt{\sqrt{\sqrt{2}}}.

It is easy to see that the equation is true. Indeed,

\sqrt{\sqrt{\sqrt{2}}}= ((2^{1/2})^{1/2})^{1/2}= 2^{\frac{1}{2^{3}}}=2^{{2}^{-3}}.

\log_{2}2^{2^{-3}}=2^{-3} and -\log_{2}2^{-3}=3.

If we were given the number 5, we would proceed in the same manner:

5=-\log_{2}\log_{2}\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{2}}}}}.

It will be seen that we have made use of the fact that the index 2 is dropped when writing the square root.

The general solution looks like this. if the given number is N, then

N=-\log_{2}\log_{2}\underbrace{\sqrt{\sqrt{\ldots \sqrt{\sqrt{2}}}}}_{N times},

the number of radical signs equalling the number of units in the given number.

More later,

Nalin Pithwa

 

Some problems for Pre-RMO practice

Below are a few problems, which I think might boost a student’s confidence for Pre-RMO.

1) Factorize (show a proof also): a^{3}+b^{3}+c^{3}-3abc

2) Bad calibration.

My bathroom scale is set incorrectly but otherwise it works fine. It shows 10 kilograms when Dan stands on it, 14 kilograms when Sarah stands on it, but 22.5 kilograms Dan and Sarah are both on it. Is the scale set too high?

3) Getting inside a brick.

You  have 3 identical rectangular bricks and a ruler. Must you use a formula, such as the Pythagorean theorem, to find the length of the brick’s diagonal?

4) In the running.

In a cross-country run, Sven placed exactly in the middle among all participants. Dan placed lower, in tenth place, and Lars placed sixteenth. Is it possible to figure out how many runners took part in the race?

5) The sands of time.

Maia bought two unusual sandglasses. One measures a nine-minute interval, and the other measures a thirteen-minute interval. A certain love potion needs to boil for exactly thirty minutes. is it possible to measure such a time interval with these sandglasses under the additional stipulation that you turn over the glass(es) for the first time just as the potion starts to boil?

6) Anyone for tennis?

Each child in a school plays either soccer or tennis. One-seventh of the soccer players also play tennis, and one ninth of the tennis players play soccer. Do more than half the children play tennis?

7) Tiresome parrot.

We recently bought a parrot. The first day it said “O” and the second day “OK”. The third day the parrot said “OKKO” and the day after that “OKKOKOOK”. If this doubling pattern continues and the parrot squawks every second, will it get through squawking on the sixteenth day?

8) A wire cube.

A 12 inch long wire is to be divided into a number of parts and from these we want to construct the frame (that is, the edges) of a cube of 1 inch on a side. Can you do it with three pieces?

9) Paper folding.

You have a strip of paper that is two thirds of a meter long. However, you need a strip exactly half a meter long. Must you have a ruler to cut off such a length?

10) No one twice as rich.

One hundred children in a school are counting their money. Each child has between 1 and 100 cents, and no child has the same amount as any other child. Is it possible to divide the children into two groups so that no child in either group will have twice as much money as any other child in the same group?

Good luck 🙂 Happy problem solving…keep on trucking…

More later…

Nalin Pithwa

Implicit differentiation example — Helga von Koch’s snowflake curve (1904)

HelgavonKoch snowflakel

Let us continue further our exploration of IITJEE Calculus. Especially, implicit differentiation.

Start with an equilateral triangle, calling it curve 1. On the middle third of each side, build an equilateral triangle pointing outward. Then, erase the interiors of the old middle thirds. Call the expanded curve curve 2. Now, put equilateral triangles, again pointing outward, on the middle thirds of  the sides of curve 2. Erase the interiors of the old middle thirds to  make curve 3. Repeat the process, as shown, to define an infinite sequence of plane curves. The limit curve of the sequence is Koch’s snowflake curve.

The snowflake curve is too rough to  have a tangent at any point. In other words, the equation F(x,y)=0 defining the curve does not define y as a differentiable function of x or x as a differentiable function of y at any point. We will encounter snowflake again when we study length.

Please download the attached figure.

More later…

Nalin Pithwa