Problem statement

Prove that any $\overrightarrow{v} \in \mathbb{R}^3$ can be approximated to arbitrary precision by an integral linear combination of unit vectors from the origin to the vertices of a regular icosahedron.

Idea

icosahedron with denotation

icosahedron with denotation

  1. $S$ = Integral linear combination of $\{O’A, O’B, O’C, O’D, O’E\}$ is dense on $\mathbb{R}^2$

  • $\{ \{n\alpha\} = n\alpha - \lfloor n\alpha \rfloor \}, n\in\mathbb{N}$ is dense on $[0,1)$

  • $\{ m\alpha + n\beta \}, m,n \in \mathbb{N},\dfrac{a}{b}\in \mathbb{R}\setminus\mathbb{Q}$ is dense on $\mathbb{R}$

  • $\overrightarrow{BE} = \overrightarrow{O’E} - \overrightarrow{O’B} \in S$, $\overrightarrow{CD} \in S$, $\|\dfrac{BE}{CD} \| \in \mathbb{R}\setminus\mathbb{Q}$

  • $2\overrightarrow{O’G} = \overrightarrow{O’B} + \overrightarrow{O’E} \in S$, $2\overrightarrow{O’H} \in S$, $\|\dfrac{O’G}{O’H} \| \in \mathbb{R}\setminus\mathbb{Q}$

    pentagon with denotation

    pentagon with denotation

  1. $T$ = Integral linear combination of $\{OA,OB,OC,OD,OE,OF\}$ is dense on $\mathbb{R}^3$

  • $\overrightarrow{OF} \in T$, $5\overrightarrow{OO’} = \overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC} + \overrightarrow{OD} + \overrightarrow{OE} \in T$, $\|\dfrac{OO’}{OF} \| \in \mathbb{R}\setminus\mathbb{Q}$

  • So you can approximate the last 2 coordinates by using the “pentagon” unit vectors, then combine it with the first axis to make the integral linear system dense on $\mathbb{R}^3$.

Aftermath

2023 is my “graduation” Putnam. I only returned to competitive math last year thanks to Dr. Dan ’s great patience (The thrill is still there like the first time.) I don’t prepare seriously enough for this year (due to the regional ICPC being pushed into fall, this deserves a blog post itself), so my performance is somewhat close to my expectation.

(More things will go into this post once the result comes out)