Can you find what the last two digits of the number $4^{1999}$ are?

Investigate $1^n + 19^n + 20^n + 51^n + 57^n + 80^n + 82^n $ and $2^n + 12^n + 31^n + 40^n + 69^n + 71^n + 85^n$ for different values of n.

What is the least square number which commences with six two's?

Largest Number

Here are the largest solutions that were sent in.

Matthew from South Farnham School:

4096, by doing 4 to the power 3 to the power 2, which is (4 x 4 x 4) x (4 x 4 x 4) = 64 x 64 = 4096.

Josh from Alameda School:

1 048 576, which is 32 to the power of 4 (32 x 32 x 32 x 32).

Greenford School, Dorset sent in 8 796 093 022 208, which is 2 to the power of 43.

Charlotte, Agnes, Susannah and Miranda from Headington Junior School offered 109 418 000 000 000 000 000, which is about 3 to the power 42.

In fact the exact value of 3⁴² is 109 418 989 131 512 359 209 can you say it aloud easily?