Problems of the Week

Here's some cool problems that I saw.

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In a square multiplication table with the numbers 1 through 9, what is the sum of all of the products within the table?

If you think about it, the sum of each column x is 1x+2x+3x+4x+5x+6x+7x+8x+9x. All of the columns are numbered 1 through 9, so by distributive property, the sum of all the numbers is
(1+2+3+4+5+6+7+8+9)*(1+2+3+4+5+6+7+8+9), or 2025. Just a cool thought.


In some languages, every consonant must be followed by a vowel. How many words can be made from the letters in the Hawaiian word 'MAKAALA' if every consonant must be followed by a vowel?

If you think about it, the letters can be separated into four separate units 'MA', 'KA', 'LA', and 'A' because each consonant must be followed by an A. You can arrange the units in 4!=24 different ways.


Let set S be the set of all positive integers that have a remainder of 12 when divided into 192. What is the median of the set?

There are two rules to the set––
1. S must be greater than 12, by definition.
2. S be a factor of 180 (because 192-12=180).
If you write out the union of the two sets above, you find the median is 36. Just simple set theory.


There are two sticks of lengths A and B, where B is greater than A. To make a triangle from these two sticks and one extra stick, the extra stick must be strictly between 8 cm and 26 cm long. What is the area of a rectangle with side lengths A and B?

This is a cool one for its geometric properties. By definition, any one of a triangle's sides can be no longer than the sum of the other sides. So, we can work it down to––
A+B<26 and B<A+8
and from solving, A and B are 9 and 17, respectively.

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We should make that a thing, where I tell you a bunch of cool problems and stuff! Fun math is the best kind of math.

Stay coolio,
John

PS.–– NEXT UP: SET THEORY!!!!!