We present two variations of the game 3-Euclid. The games involve a triplet of positive integers. Two players move alternately. In the first game, each move is to subtract a positive integer multiple of the smallest integer from one of the other integers as long as the result remains positive. In the second game, each move is to subtract a positive integer multiple of the smallest integer from the largest integer as long as the result remains positive. The player who makes the last move wins. We show that the two games have the same

The game Euclid, introduced by Cole and Davie [

Collins and Lengyel [

In this paper, we present two restrictions of Collins and Lengyel’s game. In the first restriction that we call

Recall that a position

Throughout this paper, the Sprague-Grundy value of the position

This paper is organized as follows. In Section

This paper continues our investigations of variants of Euclid and related questions; see [

We show in this section that the two games

Let

Assume by contradiction that there exists a

The

Let

every move from a position in

from every position in

Let

We will prove the existence for the game

We now present a periodicity result for the

Let

By Theorem

Conversely, assume that

Collins and Lengyel [

We now solve some special cases for

Let

By Theorem

Assume by contradiction that

We state one further result, whose proof we postpone until the end of the next section.

(i) Let

(ii) Let

In this section, we first give some basic results on the Sprague-Grundy function of the game

Let

Assume by contradiction that

Let

This is immediate from Lemmas

Let

if

if

The proof is by induction on

For (i), assume by contradiction that there exist

For (ii), assume by contradiction that there exist

Thus, by the inductive principle, the lemma is true.

Let

By the part (i) of Lemma

Let

We are now in the position to show that all results early in this section are also true for the game

The positions of Sprague-Grundy value 1 in the game

Let

there is no move from a position in

from every position that is not in

_{1}. Assume by contradiction that (i) does not hold for

In the next part, we find some connections between the

Let

We work with the game

The following corollary is the converse of Corollary

Let

By Lemma

We now prove that

Note that Corollary

We now present two theorems giving periodicity of positions of Sprague-Grundy value 1 of the forms

Let

Assume that

The previous theorem does not hold for

Let

It is sufficient to prove that the theorem holds for the game

Conversely, assume that

We now present some special classes of positions of Sprague-Grundy value 1. The following corollary follows from the above theorem by induction on

(i) Let

(ii) Let

Note that (ii) follows from (i) by Corollary

Suppose that

Conversely, suppose that

As promised at the end of the previous section, we now give the following.

For

This section extends a result from Collins and Lengyel’s work on the game 3-Euclid. Let

Let

The result holds for

For the uniqueness, assume by contradiction that there are two positive integers

For the existence, assume by contradiction that there is no integer

If

Note that each pair of these

If

This sequence contains at least

Let

By Lemma

In Theorem

Let

In this section, we examine the misère versions of two games

Let

Let

if

if

The

The positions of Sprague-Grundy value 1 in the misère version of

Some properties of periodicity of

Let

Let

Let

We now come back with the main result of this section. Note that the following theorem is also true for

The game

Consider the graph

We first assume that

either

Note that the game

The author thanks the referee for his/her helpful comments and references. He thank his supervisor, Dr Grant Cairns, for careful reading of this paper.