My Research Results on Combinatorial Game
TheoryHikaru Manabe
May 31, 2023

1

Introduction and the background of my research

In this article, I will summarize my research results for this year. In each section,
I will write the introduction and to the theory that treated and present the main
results of that study.

1.1

Combinatorial GameMy research is manily on combinatorial game theory. Combinatorial games area class of two-player games with perfect information, where players take turns
making moves, and the outcome of the game is determined solely by the positions and rules of the game. These games are characterized by the absence ofchance or hidden information.
In this paper, let the name of two players be Left and Right.

1.2

Impartial Game and Partisan Game

Impartial games are games in which the sets of options for Left and Right are
the same for each position. Impartial games have only two outcomes, P-position
and N -position that are previous player’s position and next player’s position.
partisan games are games in which the sets of options for Left and Right canbe different. For the details of partisan games, see [1],Most of my research is on impartial games, but I will also cover partisan
games at the end of this article.

1.3

Nim

One of the most important topics of combinatorial game theory is Nim. There
are few piles of stones. Each player takes their turns, and remove as manystones as she or he likes from one pile. The player who remove the last stones

1

or a stone is the winner. The complete winning theory of classical theory waspresented by C. Bouton in [2] for the first time in the history.

Figure 1.

1.4

Chocolate GameThe original chocolate games are presented in [3], and these games are mathematically the same as Nim. See Figures 2. A two-dimensional chocolate barthat I study is a rectangular array of squares in which some of the squares are
removed. A bitter square printed in black is included in some parts of the bar.
Figure3 displays an example of a two-dimensional chocolate bar. Each player
takes their turn to break the bar in a straight line along the grooves and eats
the broken piece without the bitter block. The player who manages to leave theopponent with the single bitter block (black block) is the winner.

Figure 2: Step chocolate

1.5

Figure 3: Three dimensional chocolate

Some background knowledge of combinatorial games

Here we quickly review some important concepts of combinatorial game theory.
For the detail see [4] or [5].
Definition 1.1. Let x Pand y be non-negative
Pnintegers. They are expressed innBase 2 as follows: x = i=0 xi 2i and y = i=0 yi 2i , with xi , yi ∈ {0, 1}. We
define the nim-sum x ⊕ y as follows:
x⊕y =

n
X

wi 2i ,

i=0

where wi = xi + yi (mod 2).
For impartial games without drawings, there are only two outcome classes.
Definition 1.2. (i) A position is referred to as a P-position if it is a winning
position for the previous player (the player who has just moved), as long as heplays correctly at every stage.
(ii) A position is referred to as an N -position if it is a winning position for the
next player as long as he plays correctly at every stage.
2

Definition 1.3. (i) For any position p of game G, there is a set of positions
that can be reached by precisely one move in G, which we denote as move(p).
(ii) The minimum excluded value (mex) of a set S of non-negative integers isthe smallest non-negative integer that is not in S.
(iii) Let p be the position of an impartial game. The associated Grundy number
is denoted as G(p) and is recursively defined as follows: G(p) = mex({G(h) :h ∈ move(p)}).
Definition 1.4. The disjunctive sum of the two games, which is denoted as
G + H, is a supergame in which a player may move in either G or H but not
both.
Theorem 1. Let G and H be impartial rulesets and GG and GH be the Grundy
numbers of position g played under the rules of G and position h played under
the rules of H, respectively. Thus, we have the following:(i) For any position g of G, GG (g) = 0 if and only if g is a P position.
(ii) The Grundy number of positions {g, h} in game G + H is GG (g) ⊕ GH (h).
I will now present my research results on Combinatorial Game Theory.

2

Three-Dimensional Chocolate-Bar GamesA three-dimensional chocolate bar is a three-dimensional array of cubes in whicha bitter cubic box printed in black is included in some part of the bar. Figure
4 displays an example of a three-dimensional chocolate bar. Each player takes
their turn to cut the bar on a plane that is horizontal or vertical along thegrooves, and eats the broken piece. The player who manages to leave the opponent with the single bitter cube is the winner. Examples of cut chocolate bars
are depicted in Figures 5, 6, and 7.
Example 2.1. Example of a three-dimensional chocolate bar.

Figure 4. three dimensional chocolate

3

Figure 5. Vertical cut

Example 2.2.

Figure 7.
cut

Horizontal

Figure 6. Vertical cut
Definition 2.1. Suppose f (u, v) ∈ Z≥0 for u, v ∈ Z≥0 . f is said to monotonically increase if f (u, v) ≤ f (x, z) for x, z, u, v ∈ Z≥0 with u ≤ x and v ≤ z.
We define a three-dimensional chocolate bar.
Definition 2.2. Let x, y, z ∈ Z≥0 such that y ≤ f (x, y). The three-dimensional
chocolate bar comprises a set of 1 × 1 × 1 boxes. For u, w ∈ Z≥0 such that u ≤ x
and w ≤ z, the height of the column at position (u, w) is min(f (u, w), y) + 1,
where f is a monotonically increasing function in Definition 2.1. A bitter box
exists at position (0, 0). We denote this chocolate bar by CB(f, x, y, z).
x + 1, y + 1, and z + 1 are the length, height, and width of the bar, respectively.
We present a sufficient condition for a three-dimensional chocolate bar withlength p, height q, and width r is P-position if and only if (p − 1) ⊕ (q − 1) ⊕(r − 1) = 0.
Theorem 2. For a three-dimensional chocolate game whose three coordinatesa+1
satisfy the inequality y ≤ ⌊ x+z(2m + 1) and x, z ≤ (22a+2 −k ⌋ where k = 2
a+1
2a+12 )m + 2− 1, where a, m ∈ Z≥0 The position (x,y,z) is a P-position If
and only if x ⊕ y ⊕ z = 0
This result has been already submitted to a journal, and it is now in thereview process. See [6].

3Restricted Nim with a Pass

Here I present my research result on the restricted Nim with a pass, and thisresult will be published soon as [7].
In this study, restricted Nim and restricted Nim with a pass are examined.
An interesting but difficult question in combinatorial game theory has been to
determine what happens when standard game rules are modified to allow a
one-time pass, a pass move that may be used at most once in the game and not4

from a terminal position. Once either player has used a pass, it is no longer
available. In the case of classical Nim, the introduction of the pass alters themathematical structure of the game, considerably increasing its complexity. The
effect of a pass on classical Nim remains an important open question that has
defied traditional approaches. The late mathematician David Gale offered amonetary prize to the first person to develop a solution for three-pile classicalNim with a pass.
In [8] (p. 370), Friedman and Landsberg conjectured that “solvable combinatorial games are structurally unstable to perturbations, while generic, complex
games will be structurally stable.”One way to introduce such a perturbation is
to allow a pass.
In the restricted Nim, the introduction of a pass move has a minimal impact.
There is a simple relationship between the Grundy numbers of the game and
the Grundy numbers of the game with a pass move, and the number of piles
can be any natural number.

3.1

Maximum Nim

In this section, we study maximum Nim, which is a game of restricted Nim.
Definition 3.1. If the sequence f (m) for m ∈ Z≥0 satisfies 0 ≤ f (m) − f (m −1) ≤ 1 for any natural number m, it is called a regular sequence.
Definition 3.2. Let f (m) be a regular sequence. Suppose there is a pile of nstones, and two players take turns removing stones from the pile. In each turn,the player is allowed to remove at least one stone and at most f (m) stones,
where m represents the number of stones. The player who removes the laststone is the winner. We refer to f as a rule sequence.
Here, we let f (x) = ⌈ x2 ⌉. Because 0 ≤ f (m) − f (m − 1) ≤ 1 for any m ∈ N ,f (m) for m ∈ Z≥0 is a regular sequence. Here, we examine maximum Nim ofDefinition 3.2 for f (x).
Definition 3.3. We denote the pile of m stones as (m), which we call the position of the game.
Definition 3.4. The set of all the positions that can be reached from position(t) is defined as move(t). For any t ∈ Z≥0 , we have tand v ∈ N }.
move(t) = {(t − v) : v ≤2

3.2

Three-Pile Maximum Nim

Definition 3.5. Suppose that there are three piles of stones and two players
take turns removing stones from the piles. In each turn, the player chooses apile and removes at least one stone and at most f (x) = ⌈ x2 ⌉ stones, where x
5represents the number of stones. The player who removes the last stone is thewinner. The position of the game is represented by three coordinates (s, t, u),where s, t, and u represent the numbers of stones in the first, second, and third
piles, respectively.
We can calculate the Grundy numbers of the game in Definition 3.5.
Theorem 3. Let G(t) be the Grundy number of the game in Subsection 3.1.
Then, the Grundy number G(s, t, u) of the game of Definition 3.5 satisfies thefollowing equation: G(s, t, u) = G(s) ⊕ G(t) ⊕ G(u).
Proof. This is directly from Theorem 1.

3.3

Maximum Nim with a Pass

In Subsections 3.4 and 3.5, we modify the standard rules of the games to allow
for a one-time pass, that is, a pass move that may be used at most once in thegame and not from a terminal position. Once a pass has been used by either
player, it is no longer available.

3.4

Maximum x  Nim with a Pass Whose Rule Sequence Is
f (x) = 2

The position of this game is represented by two coordinates (t, p), where t represents the number of stones in the pile. We define p = 1 if the pass is stillavailable; otherwise, p = 0.
We define move in this game.
Definition 3.6. For any t ∈ Z≥0 , we have (i) and (ii).
(i) If p = 1 and t > 0, tand v ∈ N } ∪ {(t, 0)}.
move(t, p) = {(t − v, p) : v ≤2(ii) If p = 0 or t = 0, tmove(t, p) = {(t − v, p) : u ≤and v ∈ N }.
2
Remark 3.1. Note that a pass is unavailable from position (t, 1) with t = 0,
which is the terminal position. Apparently, G(t, 0) is identical to G(t) in Section
3.1 .
According to Definitions 1.3 and 3.6, we define the Grundy number G(t, p)of the position (t, p).
p ∖ t 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 250
0 1 0 2 1 3 0 4 2 5 1
6
3
7
0
8
4
9
2 10 5 11 1 12 6 131
0 2 1 0 2 4 1 3 0 6 2 5 4 8 1 7 3 10 0 9 6 12 2 11 5 14

Figure 8: Table of Grundy numbers G(t, p)
6

The following Theorem is results of my research.
Theorem 4. Let G(s, p) be the Grundy number of position (s, p). Then, we
obtain the following:(i) G(0, 0) = 0 and G(0, 1) = 0.
(ii) For u ∈ N , if G(u, 0) = 0, then G(u, 1) = 1.
(iii) For u ∈ N , if G(u, 0) = 2, then G(u, 1) = 0.
(iv) For u, m ∈ N such that m > 1, if G(u, 0) = 2m, then G(u, 1) = 2m − 1.
(v) For u, m ∈ N , if G(u, 0) = 2m − 1, then G(u, 1) = 2m.

3.5

Three-Pile Maximum Nim with a Pass

Here, we study maximum Nim with three piles based on Definition 3.5 by modifying the standard rules of the games to allow a one-time pass. We consider
only three-pile games, although generalization to the case of an arbitrary natural
number of piles is straightforward.
We denote the position of the game with four coordinates (s, t, u, p), wheres, t, and u represent the numbers of stones in the first, second, and third piles,
respectively. Here, we define p = 1 if the pass is still available, and p = 0
otherwise.
We define a move in this game as follows.
Definition 3.7. For any s, t, u ∈ Z≥0 , we have (i) and (ii).
(i) If p = 1 and s + t + u > 0,lsm
and v ∈ N }move(s, t, u, p) = {(s − v, t, u, p) : v ≤2 t∪ {(s, t − v, u, p) : v ≤and v ∈ N }2lum∪ {(s, t, u − v, p) : v ≤and v ∈ N } ∪ {(s, t, u, 0)}.
2(ii) If p = 0 or s + t + u = 0,move(s, t, u, p) = {(s − v, t, u, p) : v ≤lsm

and v ∈ N }2 t∪ {(s, t − v, u, p) : v ≤and v ∈ N }2lum∪ {(s, t, u − v, p) : v ≤and v ∈ N }.
2

According to Definitions 1.3 and 3.7, we define the Grundy number G(s, t, u, p)of the position (s, t, u, p). Because s, t, and u represent the numbers of stonesin the first, second, and third piles, respectively, the value of G(s, t, u, p) does
not depend on the order of the arguments s, t, u.
Remark 3.2. Note that a pass is unavailable from the position (s, t, u, 1) withs + t + u = 0, which is the terminal position. It is clear that G(s, 0, 0, p),G(0, s, 0, p), and G(0, 0, s, p) are identical to G(s, p) in Section 3.4.
7

This is my main result of the research of [7].
Theorem 5. We have the following for the Grundy numbers:(i) For s ∈ Z≥0 , G(s, 0, 0, 1) = G(0, s, 0, 1) = G(0, 0, s, 1) = G(s, 1).
(ii) We suppose that s, t > 0, t, u > 0, or u, s > 0. Thus, we have the following:(ii.1) For any m ∈ Z≥0 , if G(s, t, u, 0) = 2m, then G(s, t, u, 1) = 2m + 1.
(ii.2) For any m ∈ Z≥0 , if G(s, t, u, 0) = 2m + 1, then G(s, t, u, 1) = 2m.
As easily seen, Theorem 5 gives simple.

4

Symbolic Regression to Unsolved Mathematical Problems

The result of this research is presented in [9]. This study proposes a method
to solve unsolved mathematical games using the symbolic regression library.
In mathematics, mathematicians spend time finding a formula that describes
a given data, which is often time-consuming. Here symbolic regression library
can help humans.
First, I with the help of my teachers and coaches customized a Python symbolic regression library “gplearn” by implementing new features, such as conditional branching, and a few discrete functions selected for the study. “gplearn”
uses generic programming to find a formula from data. We found that the performance of customized “gplearn” is far better than the original one. In thiscustomized “gplearn”, the person who used this library should set the conditionof branching, and we used their knowledge as mathematicians when they set
the conditions.
Secondly, they made a Swift symbolic regression library using generic programming. In this library, we implemented a new method to select the fittest
formulas by combining two methods of choosing formulas. The first method is
to choose by the smallness of the mean absolute error, and the second method
is to choose by the largeness of the percentage of the given data that can be
described without any error by a specific formula. As a result, the Swift library
can discover formulas as good as customized gplearn without using conditional
branching. Therefore, this Swift library can perform without the knowledge of
specialists in the field of research. In some examples, the Swift library could
discover fewer formulas that describe the data than the customized gelearn.
Therefore, it is undoubtedly a better library for a specific field of mathematics.
The research result shows the possibility of using generic programming inmathematics and widens the scope of the research on symbolic regression.

5

Partisan Chocolate GamesIn this study, we study Partisan Chocolate Games. This is my joint researchwith Dr.T. Abuku, Prof. R. Nowakowski, Prof. C. P. Santos and Dr. K.
Suetsugu. So far, impartial chocolate games are studied as a generalization of8

Nim. In this study, we use color blocks of chocolates and restrict the player who
can make a move to one player for each line. In general, under this rule, there
are various values. However, in the checkerboard pattern and stripe pattern,
the values are only numbers, and we found formulas to determine the values.
Definition 5.1. In the partisan game, there are two other outcome classes
beside P-position and N -position in Section 1.2. We need the following outcome
classes.
(a) A position is referred to as a L-position if it is a winning position for Left.
(b) A position is referred to as an R-position if it is a winning position for Right.

5.1

Game values

In partisan game, the study of game values is very important for analyzing
game’s outcome classes. In the following, we briefly discuss the values of thegames, but omit the description of games with special values.
Let G be a position of the game. Let GL denote the left options of G (thepositions after a left move in G), and let GR denote the right options of G (thepositions after a right move in G) as well. The game G will be denoted by thefollowing.
G = {GL | GR }In the end game, no options exist. Such a game is called an empty gameand is written as {∅ | ∅} or { | }.
Definition 5.2. For n ∈ Z, game value is defined as follows:
def(i)0 = {∅ | ∅},def(ii)n = {n − 1 | },def(iii) − n = { | −(n − 1)}.
For binaryrational numbers,
 we define the following: For j > 0 and odd m,m def m − 1 m + 1.
(iv) j =
2
2j
2j

5.2

Hackenbush

Hackenbush is a game played on a graph. Every edge is colored blue or red.
There is a special vertex called “ground” and other vertices. In her turn, Left
(resp. Right) chooses one bLue (resp. Red) edge and removes it. If deleting anedge splits the original graph into two disconnected components, the connectedcomponent that does not contain “ground” is deleted at the same time.

5.3

Rules of the Partisan Chocolate GameLet N0 be the set of all non-negative integers. We consider the chocolate game
to be the partisan version, it is called a Partisan Chocolate Game.
We define the Partisan Chocolate Game.
9

Definition 5.3. In Partisan Chocolate Game, except for the leftmost-bottom
bitter block, every block is colored blue or red.
For each column of blocks, if the top block is colored blue, Left can cut the
chocolate in two along the vertical line to its left. If the top block is colored red,
Right can cut the chocolate in two along the vertical line to its left. For eachrow of blocks, if the rightmost block is colored blue, Left can cut the chocolate in
two along the horizontal line to its bottom. If the rightmost block is colored red,
Right can cut the chocolate in two along the horizontal line to its bottom.
After cutting, the player eats the piece which does not include the bitter
block and the player who cannot cut the board is the loser as well as the original
Chocolate Game.
Note that the player can cut the board from top or right, so in positions like
Figure 9, Left does not have any legal moves.

"

B R

!

B
!

"

Figure 9: Position in which Left has no legal move
Example 6. We have a checkerboard pattern as the one in Figure 10. Left cancut the board as in Figure 12. Similarly, Right can cut the board as in Figure
13.

Figure 11: Position (3, 1)

Figure 10: Checkerboard pattern

10

Figure 12: Cut by Left

Figure 13: Cut by Right

The position or the shape of n × m chocolates is denoted by (n − 1, m − 1),so the positions of chocolates in Figures 11 and 12 are described by (1, 3) and(3, 4).

5.4

In the case of 1 × n

Every position in 1 × n Partisan Chocolate Game is the same as a position
Hackenbush string with corresponding blue and red edges. See the following
example.
Example 7. The position in Figure 14 is isomorphic to the position in Figure
15.
5

R
R

%

B

$

R

B

#

R

R

"

B

R

!

B
!

Figure 14: One-dimensional chocolate

Figure 15: Hackenbush

One can calculate the value of any position in Hackenbush string by using
a simple method. Thus, one can calculate the value of any position in 1 × n
Partisan Chocolate Game. For the details of Hackenbush string, see [5].
Definition 5.4. We define the sequence {Un : n ∈ N0 } by
 U0 = 0, 
i
n−1
X
1
U
=
−
if n > 0.

 n
2
i=0
We can calculate the value of any position in the checkerboard pattern byfollowing Theorem .
11

Theorem 8. For any x, y ∈ N0 , we have the following equations:(i) V(2x, 2y) = U2x+2y ,(ii) V(2x + 1, 2y) = V(2x, 2y + 1) = U2x+2y+1 ,(iii) V(2x + 1, 2y + 1) = U2x+2y .
For a proof, see [10].

5.5The case of n × m stripeboard pattern.

Next, we consider stripeboard pattern. The blocks of the leftmost column are
red except for the bottom bitter block, the blocks of the second column are blue,
the blocks of the third column are red, and continue alternately. See Figure 16.

···%

R B R B$R B R B#R B R BR B R B

"

B R B

!
!

"

#

$

⋯

Figure 16: Stripe-board pattern
We can calculate the value of any position in the stripeboard pattern byfollowing Theorem .
Theorem 9. For any x, y ∈ N0 , we have the following equations:

V (x, y) =










Ux−2y

(x ≥ 2y),x − 2y2(x < 2y and x is even),







 −x + 2y + 3
2(x < 2y and x is odd).

For a proof, see [10].

Acknowledgements
I would like to express my deep gratitude to Dr. Ryohei Miyadera, Dr. Koki
Suetsugu, and Dr. Tomoaki Abuku for their various guidance and advice for mystudy and this article. I would also like to thank Dr. Richard J. Nowakowski
and Dr. Carlos P. Santos very much for their collaboration in this research, andKadokawa dwango Educational institute, Academic Research Club and Keimei
Gakuin Junior and High School for their financial support.

12References[1] M. H. Albert, R. J. Nowakowski, and D. Wolfe, Lessons in play: An
introduction to combinatorial game theory, A. K. Peters (2007).
[2] C. Bouton, Nim, a game with a complete mathematical theory, Annals of
Mathematics, 3 (14): 35–39[3] A. C. Robin, A poisoned chocolate problem, Math. Gaz. 73, No. 466(1989), 341-343.
[4] A. N. Siegel, Combinatorial Game Theory, volume 146 of Graduate Studies
in Mathematics, American Mathematical Society, Providence, RI, 2013.
[5] M. H. Albert, R. J. Nowakowski and D. Wolfe, Lessons In Play: An Introduction to Combinatorial Game Theory, Second Edition, A K Peters/CRC
Press, Natick, MA., United States, 2019.
[6] H. Manabe and R. Miyadera, Previous player’s positions of threedimensional chocolate bar games, under review.
[7] R. Miyadera and H. Manabe, Restricted Nim with a Pass. To appear in
Integers.
[8] R.E. Morrison, E.J. Friedman, and A.S. Landsberg, Combinatorial gameswith a pass: A dynamic systems approach, Chaos, An Interdisciplinary
Journal of Nonlinear Science, 21 (2011), 43-108.
[9] K. Tanemura, Y. Tachibana, Y. Tokuni, H. Manabe and R.
Miyadera, Application of Generic Programming to Unsolved Mathematical Problems, Proceeding of IEEE 11th Global Conference on Consumer Electronics (GCCE), Osaka, Japan, 2022, pp. 845-849, doi:
10.1109/GCCE56475.2022.10014039.
[10] T. Abuku, H. Manabe, R. J. Nowakowski, C. P. Santos, Koki Suetsugu“Partisan Chocolate Game”, IPSJ SIG Technical Report, Vol. 2023-GI-49
No.7, pp. 1-7, 2023
[11] L. Levine, Fractal sequences and restricted Nim, Ars Combinatoria 80(2006), 113-127.

Appendix A: Publications(1) R. Miyadera and H. Manabe, Restricted Nim with a Pass. To appear in
Integers.
(2) R. Miyadera, S. Nakamura and H. Manabe, Previous Player’s Positions of
Impartial Three-Dimensional Chocolate-Bar Games. To appear in Thai
Journal of Mathematics.
13

(3) K. Tanemura, Y. Tachibana, Y. Tokuni, H. Manabe and R. Miyadera,
Application of Generic Programming to Unsolved Mathematical Problems, Proceeding of IEEE 11th Global Conference on Consumer Electronics (GCCE), Osaka, Japan, 2022, pp. 845-849(4) T. Abuku, H. Manabe, R. J. Nowakowski, C. P. Santos, Koki Suetsugu“Partisan Chocolate Game”, IPSJ SIG Technical Report, Vol. 2023-GI-49
No.7, pp. 1-7, 2023

Appendix B: Conferences(1) Y. Tokuni, R. Miyadera, H. Manabe, A. Murakami and S. Takahasi.
“Arithmetic Mean and Geometric Mean“The international conference “The24th Japan Conference on Discrete and Computational Geometry, Graphs,
and Games ”September 9-11, 2022 at Tokyo University of Science, Tokyo,
Japan.
(2) K. Tanemura, R. Miyadera, Y. Tokuni and H. Manabe. “Combinatorial Games and Genetic Programming“ The international conference “The24th Japan Conference on Discrete and Computational Geometry, Graphs,
and Games” September 9-11, 2022 at Tokyo University of Science, Tokyo,
Japan.
(3) R. Miyadera, H. Manabe and A. Singh. “P-positions of Chocolate Games“
The international conference “The 24th Japan Conference on Discrete andComputational Geometry, Graphs, and Games” September 9-11, 2022 at
Tokyo University of Science, Tokyo, Japan.
(4) R. Miyadera, H. Manabe and A. Singh. “Previous Player’s Positions ofThree-Dimensional Chocolate Games with a Restriction on the Size of
Chocolate The international conference “The 24th Japan Conference on
Discrete and Computational Geometry, Graphs, and Games” September
9-11, 2022 at Tokyo University of Science, Tokyo, Japan.
(5) K. Tanemura, Y. Tachibana, Y. Tokuni, H. Manabe and R. Miyadera,
Application of Generic Programming to Unsolved Mathematical Problems 2022 IEEE 11th Global Conference on Consumer Electronics (GCCE2022).
(6) Y. Sasaki, K. Tanemura, Y. Tokuni, R. Miyadera and H. Manabe, Application of Symbolic Regression to Unsolved Mathematical Problems,
Alliance Technology Conference-1-International Conference on Artificial
Intelligence and Applications-2023 (ICAIA), Bengaluru, India.

14

Appendix C: Awards(1) ”49th Information Processing Society of Japan” Young Researcher Award,
https://www.ipsj.or.jp/award/gi-award1.html(2) Kadokawa dwango Educational institute, Academic Research Club Special
Research Presentation Grand Prize

15