The National Young Mathematicians Award

My school entered a maths competition and (although I was in year 5 and they could enter year 6s) they chose me to enter. 3 other children entered who, like me, were in year 5. This was because the year 6s aren’t as clever.

Round 1

We went to Explore Learning and were told what the puzzle was. It seemed easy but it was actually hard. The puzzle was “You have an 11×11 grid like this:”

“And you have to cover as many tiles as you can. You have to touch the black tile once you have finished.” Easy? Think again. “You can’t go on more than 2 of the same colour tiles until you touch a different colour tile and you cannot move diagonally.” Here is an example:

We covered 116 tiles which was better than the question writers (Cambridge University). I bet you can’t beat 116.

(A fellow blogger Diaaaana, did this with 119 moves until I noticed that I had forgot to mention the “no moving diagonally” rule.  I have now added this above.  Please everyone, try again and try to beat 116).

Round 2

The question was a lot harder and we came 16th in the country. At least we were the youngest so we are probably the best mathematicians for our age.

Paradoxes

Have you heard about Zeno’s paradoxes (Achilles and the Tortoise and The Arrow Paradox). Check them out. Well I have thought of one now. It is “You have half a circle then you add half of that and then half of that and keep on going.” Here is a demonstration.

Stages 1-3Stages 4-6No matter how many times you add another half it will still never reach one. When you think of it being fractions the pattern is 1/2, 3/4, 7/8, 15/16, 31/32, 63/64 and so on. The denominator is two to the power of whatever stage and the numerator is the denominator minus one. To show it in algebra, Number of Stage=S, Numerator=N and Denominator=D. The equations are D=2^S and N=D-1. ^ means to the power of. In decimals it is 0.5, 0.75, 0.875, 0.9375, 0.96875, 0.984375. You would think Infinity times a number equals Infinity but in this case it equals one.

pi (π), circles and spheres

pi – π

pi is a number that goes on forever. So it has every number pattern possible somewhere inside it. It can be used to work out area and volume of circles and spheres.

Usually we can use pi with just 2 decimal places:

pi = 3.14

I have memorised it to 7 places as this is what a calculator shows:

pi = 3.1415927

My Dad memorised it to 21 decimal places when he was 10:

pi = 3.141592653589793238462

Circles

Circumference of a circle:

2πr (r is the radius which you can see from my picture below)

Diameter is 2 times the radius so the circumference can also be πd

Area of a circle:

πr2 (r2 means radius squared which means radius times itself)

(this picture I drew using Inkscape but I had to save it as a png as the svg file could not be seen).

Spheres

Surface area of a sphere:

4πr2

Volume of a sphere:

4/3πr3

(this picture was from wikipedia)