SOLUTION: The interiors of all unit squares of a 10 by 10 board are to be painted. Any two squares that have a side in common will be painted with a different colour of paint. What is the le

Question 1203985: The interiors of all unit squares of a 10 by 10 board are to be painted. Any two squares that have a side in common will be painted with a different colour of paint. What is the least number of colours that will be needed?
I am not satisfied with the existing solution on the Algebra.com website. It is far too simple and is not "mathematically" conveyed in any manner. Of course, this is a marvelous free service, so my feedback might be taken with distaste, and I apologise for that, but if someone can give a more mathematical solution, I would really appreciate it.
Found 2 solutions by ikleyn, MathLover1:
Answer by ikleyn(52788) (Show Source): You can put this solution on YOUR website!
.

There is a painting using two colors: it is the chessboard painting.

From the other side, one color does not produce any solution.

Hence, two colors is the least number of different colors that is needed.

It is a standard, traditional, complete and mathematically absolutely correct reasoning and a solution.

See similar solutions at this forum under these links

https://www.algebra.com/algebra/homework/word/evaluation/Evaluation_Word_Problems.faq.question.1096043.html

https://www.algebra.com/algebra/homework/Length-and-distance/Length-and-distance.faq.question.1095728.html

This problem is known for hundreds of years. With a similar solution, it is written and placed
in hundreds of textbooks in Math in different languages, that are collections of entertainment problems.
So far, I never heard that somebody would be unsatisfied or expressed doubts in validity of this solution.
In opposite, the normal human reaction is admiration that so simple and so elegant solution is possible.
You are the first person who expresses your unsatisfaction (due to unknown reason). My congratulations ( ! )

Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!

There is only one rule
Two sections that share a common edge cannot be colored the same!
Having a common corner is OK, just not an edge.
Let's start with a simple pattern like a group of nine unit squares:

How many colors do you need to color the pattern of nine squares?
You could use nine different colors, but could make do with as few as two:

it will be same for 10 by 10 board, so you will need colors

RELATED QUESTIONS
The interiors of all the unit squares of a 10 by 10 board are to be painted. Any two... (answered by greenestamps)
The interiors of all the unit squares of a 10 by 10 board are to be painted. Any two... (answered by ikleyn)
The interiors of all the unit squares of a 10 by 10 board are to be painted. Any two... (answered by ikleyn)
The interiors of all unit squares of a 10 by 10 board are to be painted. Any two squares... (answered by ikleyn)
Suppose you have a 10 by 10 checkerboard with alternating brown and white squares. If 5... (answered by stanbon)
A chess board has 32 black squares and 32 white Squares arranged in a square. How many... (answered by jim_thompson5910)
Painting a Bedroom The walls and ceiling inside your bedroom need to be painted. To save (answered by josgarithmetic)
Find the probability that a dart will (a)hit in the baseball field (b) hit in one of the... (answered by stanbon)
An open top box is to be constructed out of a 75 inch by 90 inch piece of cardboard by... (answered by ikleyn)