Is it possible to colour the naturals greater than one using two colours, red and blue, such that: the product of two red numbers is blue, and the product of two blue numbers is red?

I don’t think this is possible. Suppose it is possible. Consider the number 16=2*2*2*2. Without loss of generality we colour 2 red. Then 16 is red being the product of two 4’s and 4 is blue being the product of two 2’s. That is, 4= 2*2=red*red=blue and 16= 4*4=blue*blue=red. We may also understand 16 as (4*2)*2=(blue*red)*red. If we define blue*red as red then we reach a contradiction. If we define blue*red as blue then we also reach a contradiction. Cool puzzle.

There is a way to write it up without saying: “If we define blue*red as red” or “If we define blue*red as blue” which makes the solution look a lot nicer. We don’t know anything about red*blue, so it is undesirable to give it any value. The right way to go is: “If eight is red” or “If eight is blue”.

Alexsaid, on 2013/05/26 at 04:14I don’t think this is possible. Suppose it is possible. Consider the number 16=2*2*2*2. Without loss of generality we colour 2 red. Then 16 is red being the product of two 4’s and 4 is blue being the product of two 2’s. That is, 4= 2*2=red*red=blue and 16= 4*4=blue*blue=red. We may also understand 16 as (4*2)*2=(blue*red)*red. If we define blue*red as red then we reach a contradiction. If we define blue*red as blue then we also reach a contradiction. Cool puzzle.

pgadeysaid, on 2013/05/28 at 16:06Yup — Good analysis of the problem!

There is a way to write it up without saying: “If we define blue*red as red” or “If we define blue*red as blue” which makes the solution look a lot nicer. We don’t know anything about red*blue, so it is undesirable to give it any value. The right way to go is: “If eight is red” or “If eight is blue”.