eon

ecstatic over numbers

To the fourth dimension …

Posted by tpc at April 1st, 2010

and beyond? xkcd mentioned this game Miegakure about puzzle solving in four dimensions. I have my doubts about this but it’ll certainly be interesting.

Posted in Fun Stuff, Problems| No Comments |

Another neat number puzzle

Posted by tpc at April 8th, 2009

I chanced upon this neat number puzzle devised by William Wallace, via wild about maths. It’s form is certainly recognizable, but still the effort deserves praise.

I remember a similar puzzle played with a deck of cards. You lay out the cards in rows and ask the participant to tell who which row his card is in. The trick is to collect the cards back horizontally, but deal out the next four rows vertically, this way after a few deals, you can tell which card is it.

This happen to tie in with a class I’m teaching tomorrow, maybe I’ll print out the cards and try it in class.

Posted in Fun Stuff, Number Theory, Problems| No Comments |

Separating the variables

Posted by tpc at January 13th, 2009

We should all be familiar with the method of separation of variables for first order ordinary differential equations. Here’s a neat example from John Starrett’s article in Amer. Math. Monthly.

$\displaystyle \frac{dy}{dx} = \frac{ y^3 + x^2y -x -y}{ x^3+ xy^2 -x+y}$
(more…)

Posted in Calculus/Analysis, Problems| No Comments |

Einstein’s Puzzle

Posted by tpc at May 6th, 2008

I’ve been looking at logic puzzles as a break from all the frenzied examination related activities. An internet search will inevitably throw up the old nugget known as Einstein’s IQ quiz or puzzle. The wiki entry has a variant called the Zebra puzzle and at least a proper reference.

(more…)

Posted in Fun Stuff, Problems| 1 Comment |

Solving Mathematical Problems

Posted by tpc at June 19th, 2007

a personal perspective by Terence Tao. This is a new edition of a book which was written by Tao more than 15 years ago, which means when he was only 15! It’s a thin little book that takes a leisurely look at solving some competition type problems. The coverage is not huge, but the author take pains to go through in great detail various strategies one can adopt in solving problems. Quite a nice book but very pricey for 102 pages.

I found exercise 2.1 quite fun.

In a parlour game, the ‘magician’ asks one of the participants to think of a three-digit number $abc_{10}$ . Then the magician asks the participant to add the five numbers $acb_{10}, bac_{10}, bca_{10}, cab_{10}$ and $cba_{10}$ , and reveal their sum. Suppose the sum was 3194. What was $abc_{10}$ ?

My solution is this. If we add all the six permutations, we know that the sum equals
$(2a+2b+2c) \times 100 + (2a+2b+2c) \times 10 + (2a+2b+2c)$
$= (a+b+c) \times 222$ .
So we just need to know the multiples, $1 \times 222, \ldots, 27 \times 222$ . Take the smallest multiple larger than the given number, and check by subtracting the difference and summing the digits. You do not have to do it with more than 5 different multiples.

$15 \times 222 = 3330; 3330 - 3194 = 136$ .
But 1+3+6 = 10 , so incorrect.
now $16 \times 222 -3194 = 136 + 222 = 358$ .
And 3+5+8 = 16 and we found our number.

Posted in Books, Problems| 1 Comment |

Trigonometric problem

Posted by tpc at April 30th, 2007

Here’s my solution to a nice little trigonometric problem posted by miss loi.
Show that
$\frac{ \tan x + \sec x - 1} { \tan x - \sec x +1 } \equiv \tan x + \sec x$

$\frac{ \tan x + \sec x - 1} { \tan x - \sec x +1 } = \frac{ \sin x + 1 - \cos x } { \sin x - 1 + \cos x }$
$= \frac{ \sin x + 1 - \cos x } { \sin x - 1 + \cos x } \times \frac{ \sin x + 1 + \cos x } { \sin x + \cos x + 1 } = \frac{ (\sin x + 1)^2 - \cos^2 x } { (\sin x + \cos x)^2 - 1 }$
$= \frac{ \sin^2 x + 2 \sin x +1 - \cos^2 x } { 2 \sin x \cos x } = \frac{ 2 \sin^2 x + 2 \sin x } { 2 \sin x \cos x }$
$= \tan x + \sec x$
(QED)

Posted in Problems| 3 Comments |

Functional Equations

Posted by tpc at March 22nd, 2007

I spotted this little book at the library and flipped through it. I saw the names Cauchy, Euler and d’Alembert and was immediately thinking “oh differential equations” until I flipped to this lovely nested square root
$\sqrt{ 1+2 \sqrt{1+3\sqrt{1+4\sqrt{\ldots}}}}$
courtesy of Ramanujan.

This book turns out to be a guide to competition problem solving by Christopher Small. I’m not very big on competition problem solving, probably because I can’t do most of them, but I do like to work on some of them every once in a while. I’ve read most of the book, and quite like it.

Oh, the expression above evaluates to 3. Try it!

Posted in Books, Problems| No Comments |

Recurring Decimals

Posted by tpc at September 2nd, 2006

Came across a funny little problem last week. Explain the following:

For primes $p \ge 7$ , the decimal expansion of $\frac{1}{p}$ is non-terminating. Is there a formula for the number of repeating digits?

$\frac{1}{7} = 0.\overline{142857}$ 6 digits

$\frac{1}{11} = 0.\overline{09}$ 2 digits

$\frac{1}{13} = 0.\overline{076923}$ 6 digits

$\frac{1}{17} = 0.\overline{0588235294117647}$ 16 digits

and my favourite twin primes

$\frac{1}{41} = 0.\overline{02439}$ 5 digits

$\frac{1}{43} = 0.\overline{023255813953488372093}$ 21 digits

Posted in Number Theory, Problems| 2 Comments |

Integration problem from pAt84

Posted by tpc at December 27th, 2005

My latexrender (installed by my wife with much pain) has gone to disuse. So I’m putting up this problem posted by pAt84.

Version 1
$\mbox{\Large \mu = \int_0^1 \frac{ \big( \sum_{j=0}^n a_j t^j \big) \big( \sum_{i=1}^n i b_i t^i \big) } { \sqrt{\big( \sum_{j=1}^n j a_j t^j \big) ^2 + \big( \sum_{i=1}^n i b_i t^i \big)^2 }} dt}$

Version 2
$\mbox{\Large \mu = \int \frac{p(x) q\prime (x)}{ \sqrt{ p\prime (x)^2 + q \prime (x)^2} } dx}$

Posted in Calculus/Analysis, Problems| 1 Comment |

A Calendar Puzzle

Posted by tpc at December 25th, 2005

A simple little puzzle. Suppose you are given two ordinary 6 sided dice. Is it possible to put the numbers 0-9 (with repetition) onto the faces of both dice, such that using both dice you can display all the days of the month i.e. 01 - 31 .

Posted in Combinatorics, Fun Stuff, Problems| 9 Comments |

Elementary Proofs

Posted by tpc at February 8th, 2005

No one shall expel us from the paradise that Calculus has created.
- tpc

The fact that an analytic solution to number theoretical problems exist is bewildering. And often, the analytic proof is much simpler than the elementary proof. The Prime Number Theorem is the most famous example. Another is the evaluation of $\zeta(2)$ and $\zeta(3)$ in terms of double integrals.

The following is strictly not a number theory problem:
A rectangle is divided into smaller rectangles. Each of the smaller rectangles has the property that at least one of the sides has integer length. Show that the large rectangle has the same property.

I first encountered the problem at kuro5hin via Chapterzero, but apparently it dates back to de Bruijin in the sixties. I couldn’t solve it but as luck would have it, saw a solution in Paul Zeitz’s The Art and Craft of Problem Solving, which provided a reference to an article in American Math Monthly (vol 94) by Stan Wagon which gave 14 proofs, including a one line calculus proof.

Posted in Calculus/Analysis, Number Theory, Problems| 2 Comments |

Simple Problem

Posted by tpc at February 7th, 2005

A little number theory problem posted by higherpi.
Let n>1 be an integer such 2^n + n^2 is prime.
Prove that $n \equiv 3 \pmod{6}$ .

(more…)

Posted in Number Theory, Problems| 5 Comments |

Maple vs Excel

Posted by tpc at January 16th, 2005

AIME 1994
Find n for which
$\lfloor \log_2 1 \rfloor + \lfloor \log_2 2 \rfloor + \lfloor \log_2 3 \rfloor + \ldots + \lfloor \log_2 n \rfloor = 1994$ .

A fairly straightforward problem that can be worked out quickly once you see the pattern. 313 was my initial answer which was wrong. (The actual answer is 312.) The funny thing is that I first tried to check it with Maple and it returned an answer that was slightly wrong. My guess is bad code resulting in some round off error.

So I checked with Excel and found the answer and my initial error. Athough admitting that I like a microsoft product will destroy whatever “geek” status I that possess, I have to say that there are few things you cannot do with Excel. It is also an excellent (pun not intended) way to investigate patterns in number theory.

Posted in Problems| 1 Comment |

Monks and Fixed Points

Posted by tpc at December 10th, 2004

A monk starts to climb a mountain at 8:00 am and reaches the summit at noon. He spends the rest of the day and that night on the summit. The next morning he leaves the summit at 8:00 am and descends by the same route he used the day before, reaching the bottom at noon. Prove that there is time between 8:00 am and noon at which the monk was at exactly the same spot on the mountain on both days. Note that the monk can walk at different speeds, rest, or even go backward whenever he wants.

A nice problem. Source unknown.

Posted in Problems| 5 Comments |

Irritating Integral

Posted by tpc at November 28th, 2004

It took me almost a week to prove $\int_0^{\pi} \log (1+4\cos^2(u)) du = 2\pi \log{\phi}$ where $\phi = (\sqrt{5}+1)/2$ .

A time consuming but fruitful exercise. I had tried all the integration tricks I know, resorted to computer packages Maple and Mathematica - Integrator . In the end, I used contour integration and it took another two days to find the correct contour. I may not publish my solution here, in case I meet some smug kid who badly need to be tormented.

I found the problem here, the given solution is a heuristic argument.