How to deal with failure?

If you are an average student (like me…) and are part of an insane curriculum (like having 5 advanced math courses in one semester) you may fail in few of your courses at college, even after trying your best to pass them.  In such situations you may be able to control your panic by reading this Reddit post:

I think I am going to fail Calc II. What can I do better?
byu/Thaw6663 inmath

I will suggest you to go through the comments of that post.

Nim

Nim is a very old game with precise mathematical theory, and one player can always force a win.

The game is Nim is played as follows: Any number of matches/pebbles are arranged in heaps, the number of heaps and the number matches/pebbles in each heap, being arbitrary.  There are two players, A and B. The first player A takes any number of matches/pebbles from a heap, he/she may take only one or any number up to the whole of the heap, but he/she must touch one heap only. B then makes a move conditioned similarly, and the players continue to take alternate turns of picking matches/pebbles. The player who takes the last match/pebble wins the game.

We define a winning position as a position such that if one player P (A or B) can secure it by his/her move, leaving his/her opponent Q (B or A) to move next, then, whatever Q may do, P can play so as to win the game. Any other position we call a losing position.

Next, we express the number of matches in each heap in the binary scale and form a figure by writing down one under the other. Then we add up the columns. For example, consider the following position:

Then, (1,3,5,7) position gives the following figure:

001
011
101
111
—
224

If the sum of each column is even (which is the case above), then the position is correct. An incorrect position is one which is not correct, thus (1,3,4) is incorrect.

Then we have the following result:

A position in Nim is a winning position if and only if it is correct.

For the proof/discussion/variations of rules, see § 9.8, of G. H. Hardy and E. M. Wright’s An Introduction to the Theory of Numbers.

But designing an elaborate winning strategy, i.e. ensuring that you always stay in winning position, is not so easy (though we know it exists!). For example, watch this video by Matt Parker:

Finite Sum & Divisibility – 2

Earlier this year I discussed a finite analogue of the harmonic sum. Today I wish to discuss a simple fact about finite harmonic sums.

If is a prime integer, the numerator of the fraction is divisible by .

We wish to treat the given finite sum modulo , hence we can’t just add up fractions. We will have to consider each fraction as inverse of an integer modulo p. Observe that for , we have inverse of each element in the multiplicative group , i.e. there exist an such that .

For example, for , we have and .

Hence we have

for all .

Hence we have:

Thus we have:

The desired result follows by summation.

We can, in fact, prove that the above harmonic sum is divisible by , see section 7.8 of G. H. Hardy and E. M. Wright’s An Introduction to the Theory of Numbers for the proof.

Farey Dissection

The Farey sequence of order is the ascending sequence of irreducible fractions between 0 and 1, whose denominators do not exceed . This sequence was discovered by Charles Haros in 1806, but Augustin-Louis Cauchy named it after geologist John Farey.

Thus belongs to  if and , the numbers 0 and 1 are included in the forms and . For example,

Following are the  characteristic properties of Farey sequence (for proofs refer §3.3, §3.4 and §3.7 of G. H. Hardy and E. M. Wright’s An Introduction to the Theory of Numbers):

1. If and  are two successive terms of , then .
2. If ,  and  are three successive terms of , then .
3. If and  are two successive terms of , then the mediant  of  and  falls in the interval .
4. If , then no two successive terms of  have the same denominator.

The Stern-Brocot tree, which we saw earlier while understanding the working of clocks, is a data structure showing how the sequence is built up from 0 (=0/1) and 1 (=1/1) by taking successive mediants.

Now, consider a circle of unit circumference, and an arbitrary point of the circumference as the representative of 0 (zero), and represent a real number by the point whose distance from , measured round the circumference in the anti-clockwise direction, is .

Plainly all integers are represented by the same point , and numbers which differ by an integer have the same representative point.

Now we will divide the circumference of the circle in the following manner:

1. We take the Farey sequence , and form all the mediants of the successive pairs , . The first and last mediants are and . The mediants naturally do not belong themselves to .
2. We now represent each mediant by the point . The circle is thus divided up into arcs which we call Farey arcs, each bounded by two points and containing  one Farey point, the representative of a term of . Thus is a Farey arc containing the one Farey point .

The aggregate of Farey arcs is called Farey dissection of the circle. For example, the sequence of mediants for , say is

And hence the Farey disscetion looks like:

Let . If is a Farey point, and, are the terms of which precede and follow , then the Farey arc around  is composed of two parts, whose lengths are

respectively. Now , since and are unequal (using the point (4.) stated above)and neither exceeds ; and (using the point (3.) stated above). We thus obtain:

Theorem: In the Farey dissection of order , there , each part of the arc which contains the representative has a length between and .

For example, for we have:

Using the above result, one can prove the following result about rational approximations (for more discussion, see §6.2 of  Niven-Zuckerman-Montgomery’s An Introduction to the Theory of Numbers):

Theorem: If is a real number, and a positive integer, then there is an irriducible fraction such that and

One can construct a geometric proof of Kronceker’s theorem in one dimension using this concept of Farey dissection. See §23.2 of G. H. Hardy and E. M. Wright’s An Introduction to the Theory of Numbers  for details.