On a combination of the cyclic Nimhoff and subtraction games

Abstract

In this paper, we study a combination (called the generalized cyclic Nimhoff) of the cyclic Nimhoff and subtraction games. We give the G-value of the game when all the G-value sequence of subtraction games have a common h-stair structure. 1

Some extensions of Delete Nim

“…(3) Since k s − k can be divided by k(k − 1), the reminder of k s divided by k(k − 1) is k. Thus, the largest k-evenoid number less than k s is k s − k. Therefore, for the case s ≤ 2, we have proved in (2).…”

“…In this section, we review the rules of Delete Nim and the determination of winners and losers as previously mentioned in Abuku and Suetsugu [2].…”

“…Theorem 2.2 (Abuku and Suetsugu [2]). Let x, y be a Delete Nim position, where x and y represent the number of tokens in each heap.…”

“…How to split the heap into n heaps at the last of the proof will be presented in Lemma 5.5 (2). 4 No-more-than-half-delete Nim Next, we introduce No-more-than-half-delete Nim or NMTH-delete Nim.…”