  Yama Nim is a two heaps Nim game introduced in the second author's Master
Thesis, where the player takes more than $2$ tokens from one heap, and return
$1$ token to the other heap. Triangular Nim is a generalization, where the
player takes several tokens from one heap, and return some tokens (at least one
token) to the other heap, so that the total number of the tokens in the heaps
decrease strictly. In this paper, we investigate their winning strategies,
Grundy numbers, and their variations and generalizations. Particularly
interesting is the Wythoff variations, where in addition to the Yama/Triangular
Nim moves, the player is allowed to take tokens from both heaps, say $i$ tokens
from the first heap and $j$ tokens from the other, under some restrictions for
$i$ and $j$. For example when we force $i=j>0$ for the Triangular Nim, then the
pair of non-negative integers $(x, y)$ with $x$ less than or equal to $y$ is in
the $P$-position if and only if $(x, y) is in {(0, 0), (0, 1), (1, 3), (3, 6),
(6, 10), (10, 15), ...}$, namely the winning strategy is described by
triangular numbers. In other rulesets, we also found examples where the square
numbers, pentagonal numbers, geometric progressions, and so on.
