MathDB

Adel draws an

m \times n

grid of dots on the coordinate plane, at the points of integer coordinates

(a,b)

where

1 \le a \le m

and

1 \le b \le n

. He proceeds to draw a closed path along

k

of these dots,

(a_1, b_1)

(a_2,b_2)

,...,

(a_k,b_k)

, such that

(a_i,b_i)

and

(a_{i+1}, b_{i+1})

(where

(a_{k+1}, b_{k+1}) = (a_1, b_1)

) are

1

unit apart for each

1 \le i \le k

. Adel makes sure his path does not cross itself, that is, the

k

dots are distinct. Find, with proof, the maximum possible value of

k

in terms of

m

and

n

Problem Statement