# JOMO 6, Long 3

A teacher writes down three numbers, 1, 2 and 3, on the whiteboard. Now, every student take turns to the whiteboard and erase one number, and then replace it by the sum of the two numbers left. After some turns, is it possible to have the numbers: $$6^{2012}, 7^{2013}, 8^{2014}$$ on the whiteboard at the same time? Give proof.

Note by Yan Yau Cheng
4 years ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $$\boxed{123}$$

Sort by:

Notice that a curious characteristic (even/uneven) of the initial sum of numbers never changes. We have that $$1+2+3$$ is even, but $$6^{2012} + 7^{2013} + 8^{2014}$$ is uneven. By the Invariance Principle, it is not possible to have these numbers on the whiteboard. $$\boxed{\mathbb{QED}.}$$

- 4 years ago

After every turn, the sum of the numbers on the whiteboard will be even (from a,b,c on the whiteboard we will gen 2*(a+b) or the analogs) Since 6^2012+7^2013+8^2014 is odd the answer is NO

- 2 years, 5 months ago