MathDB

Mark the Martian and Bark the Bartian live on planet Blok, in the year

2019

. Mark and Bark decide to play a game on a

10 \times 10

grid of cells. First, Mark randomly generates a subset

S

\{1, 2, \dots, 2019\}

with

|S|=100

. Then, Bark writes each of the

100

integers in a different cell of the

10 \times 10

grid. Afterwards, Bark constructs a solid out of this grid in the following way: for each grid cell, if the number written on it is

n

, then she stacks

n

1 \times 1 \times 1

blocks on top of one other in that cell. Let

B

be the largest possible surface area of the resulting solid, including the bottom of the solid, over all possible ways Bark could have inserted the

100

integers into the grid of cells. Find the expected value of

B

over all possible sets

S

Mark could have generated.
Proposed by Yang Liu

Problem Statement