Puzzle Resources

Just a few collections of good puzzles and exercises.

Putnam Directory has past Putnam exams and practice problems.

Project Euler has many dynamic programming problems with a mathematical twist.

Math.StackExchange has a few puzzle posts.

M. V. Sapir’s “Some group theory problems” arXiv:0704.2899 [math.GR]

Programming or Combinatorial Puzzles

Reddit’s Programming Challenges.

ACM’s International Collegiate Programming Contest has some fascinating problems.

Math.SE Puzzles

1 [Math.SE]. Evaluate the following continued fraction:
\displaystyle x = 1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{2+ \cdots}}}}}

2 [Math.SE]. Calculate \displaystyle \int_0^1 \int_0^1 \frac{ \mathrm{d}x \, \mathrm{d}y}{1+xy+x^2y^2}

3 [Math.SE]. Compute \displaystyle \int_{y=0}^{1} \int_{x=y}^{1} \frac{x^2}{y^2} e^{\frac{-x^2}{y}}\,\mathrm{d}x\,\mathrm{d}y.

4 [Math.SE]. Evaluate \displaystyle\int_{y=-2}^{2} \int_{x=0}^{\sqrt{4-y^2}} x e^{{(4-x^{2})}^{3/2}} \mathrm{d}x\,\mathrm{d}y. [Hint: swap the order of integration!]

5 [Math.SE]. Compute \displaystyle \int^{4}_{0} \int^{2}_{\sqrt {x}} \frac{x^{2} e^{y^{2}}}{y^{5}}\mathrm{d}y\,\mathrm{d}x.

6 [Math.SE]. Let -1<\alpha,\beta<1. Define
\displaystyle f(\alpha,\beta)=\int^{\infty}_{0}\frac{x^{\alpha}}{1+2x\cos(\pi\beta)+x^{2}}\mathrm{d}x.
Prove the equality
\displaystyle f(\alpha,\beta)=\frac{\pi\sin(\pi\alpha\beta)}{\sin(\pi\alpha)\sin(\pi\beta)}

holds.

7. Compute \displaystyle \int^{1}_{0}x^{x}(1-x)^{1-x}\sin(\pi x)\,\mathrm{d}x.

In one’s life there are levels in the pursuit of study.

In the lowest level, a person studies but nothing comes of it, and he feels that both he and others are unskillful. At this point he is worthless.

In the middle level he is still useless but is aware of his own insufficiencies and can also see the insufficiencies of others.

In a higher level he has pride concerning his own ability, rejoices in praise from others, and laments the lack of ability in his fellows. This man has worth.

In the highest level a man has the look of knowing nothing. These are the levels in general.

But there is one transcending level, and this is the most excellent of all. This person is aware of the endlessness of entering deeply into a certain way and never thinks of himself as having finished. He truly knows his own insufficiencies and never in his whole life thinks that he has succeeded. He has no thoughts of pride but with self-abasement knows the way to the end.

It is said that Master Yagyu once remarked, “I do not know the way to defeat others, but the way to defeat myself.”

Throughout your life advance daily, becoming more skillful than yesterday, more skillful than today. This is never-ending.

Hagakure, Chapter 1

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