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7
4
,
11
5
,
7
3
)
remains open. For all instances
of the game in which the solution has been found, the hider places the two objects
in such a way that once the searcher finds one object, the remaining object is placed
equiprobably in one of the four directions, as far away as possible. Even though the
amount of evidence is not overwhelming, there may be an underlying principle:
The solution of the game for
h
in
[
2
)
∪
[
A Kikuta-Ruckle Conjecture for Caching Games
Let
denote the
caching game in which the hider can dig in
n
directions for a total length of
one unit, hiding
k
objects, of which the searcher has to retrieve
j
and he can
dig a total length
h
. Then the hider places the objects in such a way that once
the searcher finds a single object at distance
x
, then the remaining objects are
optimally placed in the remaining game
Γ
(
j
,
k
,
n
,
h
)
h
x
1
−
x
)
−
Γ
(
j
−
1
,
k
−
1
,
n
,
.
If such a recursive principle exists, it should also apply to the Kikuta-Ruckle
conjecture that we exhibited in the first section, and other versions of that conjecture
which can be found in the papers by Kikuta and Ruckle on accumulation games.
References
1.
N. Alon, P. Frankl, H. Huang, V. Rödl, A. Ruci nski, B. Sudakov, Large matchings in uniform
hypergraphs and the conjectures of Erdös and Samuels, J. Combin.Theory, Series A,
119
,
1200-1215, (2012)
2.
S.
Alpern,
R.
Fokkink,
K.
Kikuta,
On
Ruckle's
conjecture
on
accumulation
games,
SIAM J. Control Optim.
48
no 8, 5073-5083, (2010)
3. S. Alpern, R. Fokkink, J. op den Kelder, T. Lidbetter, Disperse or unite: a mathematical model
for coordinated attack, Decision and Game Theory for Security (GAMESEC2010), LNCS
6442, 221-233, (2010)
4. S. Alpern, R. Fokkink, C. Pelekis, A solution of the Kikuta-Ruckle conjecture on cyclic
caching of resources, J. Optim. Theory Appl.
153
no 3, 650-661, (2012)
5. N. Biggs, Some odd graph theory, Annals New York Academy of Sciences
319
no 1, 71-81
(1979)
6. P. Erdös, A problem on independent r-tuples, Ann. Univ. Sci. Budapest Eötvös Sect. Math.
8
,
93-95, (1965)
7. K. Jogdeo, S.M. Samuels, Monotone convergence of binomial probabilities and a generaliza-
tion of Ramanujan's equation, Ann. Math. Statistics
39
no 4, 1191-1195, (1968)
8. J. op den Kelder, Disperse or unite: a mathematical model for coordinated attack, TU Delft
9. K. Kikuta, W. Ruckle, Accumulation games, Part 1: noisy search, J. Optim. Theory Appl.
94
no 2, 395-408, (1997)
10.
K. Kikuta, W. Ruckle, Continuous accumulation games in continuous regions, J. Optim.
Theory Appl.
106
no 3, 581-601, (2000)
11.
K. Kikuta, W. Ruckle, Continuous accumulation games on discrete locations, Naval Res.
Logistics,
49
no 1, 60-77, (2002)
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