Coming up with Problems

I was just wondering; how do brilliant challenge masters come up with these problems?

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

There are several ways that we create the problems.

1) Playing around with stuff - This problem arose simply because the fraction (function) was something interesting to look at and understand. I then had to find a way to turn it into a problem.

2) Work from a certain principle that we want you to apply - For example, if we want to test that you know how to use Pythagoras theorem, we can simply give you a right angled triangle with 2 sides and ask for the hypotenuse. At higher levels, we may try and disguise the right angle (say radius and tangent of circle), or even remove it completely. This problem on telescoping series is an old chestnut (and I can't honestly say that it was an actual creation).

3) Being familiar with various techniques and 'tricks' - Most basic problems are created to ensure students are aware of a variety of approaches, especially if these are not dealt with in school. This problem arises naturally if you are aware of how to square a multinomial and take care of cross terms.

4) Combining different techniques - For example, basic Permutations and Combinations questions can get boring / repetitive after a while, and so we combine it with Principle of Inclusion and Exclusion, or Double-counting. This problem arose from combining Remainder-Factor Theorem with Factorization of Integers.

5) Knowing a lot more math - Some of these problems are a special case, or a simplified version of a difficult approach / technique. However, such questions are rare, as the aim isn't for you to simply learn more math, but to apply what you already know. We will ensure that such problems have an elementary approach, and will talk about the bigger picture in the solutions. This problem arose because I knew the determinant of a matrix with that general form.

Calvin Lin Staff - 8 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$