Friend’s kid is a happy, bright boy attending junior high. Last weekend, he showed me some of his math home work questions while we chatted online. One of them was interesting enough that I thought it is worth to share with everyone who reads my blog.
The question was simple enough: Find a the value of ? in the given Hexgon.
With a degree in applied science, this was a non-brainer to me.
“I found the pattern of these numbers. The formula is (next = 3*previous + 1).” I proudly announced after some quick calculation. The young man's face brighten with admiration and I was content with my own mathematical ingenuity.
I was still thinking of this question after got offline. What IS the answer then? If my pattern was correct, then the value should be either 1(because 1*3+1=4) or 364*3 + 1 = 1093. So this question should have two answers and both of them are correct. I was satisfied and quickly forgot about this question.
For some strange reason, the shape of this hexagon jumped into my head while I was preparing dinner today. It can’t be right! Both of these answers violate a basic geometrical rule—in any geometrical figure, the length of the longest side should be less than the sum of lengths of all other sides. This sounds dry and boring, but it is actually pretty simple. In our example, if we took 1 as the answer. When we sum the number from 1 to 121, we get 1+4+13+40+121 = 179. The longest side is 364, which is greater than 179. So, we CANNOT form a hexagon with these values. Same conclusion can be drawn if we take 1093 as the answer.
I tried to prove myself correct by using sticks with given lengths. But no matter what unit I use, the sticks I found are either too short or too long. However, it is easy to test with sticks with more reasonable measurements. Find 6 sticks with length of 5cm, 6cm, 7cm, 8cm, 9cm, and 50cm respectively. Try to form a hexagon with them and you will know what I mean. (Hint: there are lots of falling twigs outside after the last night’s storm in Vancouver).