# Stuck on Gelfand Algebra problem 43, please help!

I'm self studying mathematics for leisure and using Gelfand's Algebra. He poses problem 43 and proves it right afterwards (see image below).

I've been struggling to understand the "proof". I understand some parts, like: the ends won't have marks, and that the same color marks cannot be on the same fractional $$1/20$$ piece.

However, the main argument eludes me. Please help, thanks! ## 1 Answer

The crux of the argument is that $$(k+l)/20$$ is between $$k/7$$ and $$l/13$$. That is $$\frac k7<\frac{k+l}{20}<\frac l{13}$$ whenever $$k/7 and $$\frac k7>\frac{k+l}{20}>\frac l{13}$$ whenever $$k/7>l/13$$. This boils down to proving that $$\frac k7-\frac{k+l}{20}\qquad\text{and}\qquad \frac{k+l}{20}-\frac l{13}$$ have the same sign. I would suggest simplifying these.

• Thanks for your help, perhaps I'm not envisioning the problem correctly. I can't figure out what "k+l/20" means. This is the way I am seeing the problem: imgur.com/a/gxwyrm1 – Asif Kazmi Dec 29 '18 at 15:59
• @AsifKazmi $(k+l)/20$ is just a fraction with 20 as the denominator. The stick has markings at the points $k/7$ for $k=1,2,3,4,5,6$, and at $l/13$ for $l=1,2,3,4,5,6,7,8,9,10,11,12$. The stick is cut at $1/20,\,2/20,\,3/20,\,\dots\,,\,19/20$. Here, you are showing that (say) $1/7 < 3/20<2/13$. I.e. $(k+l)/20$ is between $k/7$ and $l/13$ for all the possible values of $k,l$. – John Doe Dec 29 '18 at 16:04