---
tag: 8HKFHQPX13
kind: statement
role: proposition
name: Formula for cold positions
sources:
  - wikipedia: en:10222175@r1338739976
    locator: Wythoff's game, section Formula for cold positions
    note: adopted
    mention: wikipedia://en/10222175#BF5Z
    sections: [Statement]
inventory: [P1XEMQS1GM]
---

## Statement

Wythoff discovered that the cold positions follow a regular pattern determined by the [[golden ratio|0SC8T7548D]]. Specifically, if *k* is any natural number and

$$n_k = \lfloor k \phi \rfloor = \lfloor m_k \phi \rfloor -m_k \,$$

$$m_k = \lfloor k \phi^2 \rfloor = \lceil n_k \phi \rceil = n_k + k \,$$

where φ is the golden ratio and we are using the floor function, then (*n*<sub>*k*</sub>, *m*<sub>*k*</sub>) is the *k*<sup>th</sup> cold position. These two sequences of numbers are recorded in the Online Encyclopedia of Integer Sequences as and , respectively.

The two sequences *n<sub>k</sub>* and *m<sub>k</sub>* are the Beatty sequences associated with the equation

$$\frac{1}{\phi} + \frac{1}{\phi^2} = 1 \,.$$

As is true in general for pairs of Beatty sequences, these two sequences are [[complementary|0RKF1ZT5JJ]]: each positive integer appears exactly once in either sequence.