MathDB

The numbers from

1

100

are placed without repetition in each cell of a

10 \times 10

board. An increasing path of length

k

on this board is a sequence of cells

c_1, c_2, \ldots, c_k

such that, for each

i = 2, 3, \ldots, k

, the following properties are satisfied:
• The cells

c_i

and

c_{i-1}

share a side or a vertex; • The number in

c_i

is greater than the number in

c_{i-1}

.
What is the largest positive integer

k

for which we can always find an increasing path of length

k

, regardless of how the numbers from 1 to 100 are arranged on the board?

Problem Statement