Answer to Question #149828 in Combinatorics | Number Theory for Wesam

Question

On a 9×9 square lake composed of unit squares, there is a 2×4 rectangular iceberg also composed of unit squares (it could be in either orientation; that is, it could be 4×2 as well). The sides of the iceberg are parallel to the sides of the lake. Also, the iceberg is invisible. Lily is trying to sink the iceberg by firing missiles through the lake. Each missile fires through a row or column, destroying anything that lies in its row or column. In particular, if Lily hits the iceberg with any missile, she succeeds. Lily has bought missiles and will fire all of them at once. Let N be the smallest possible value of n such that Lily can guarantee that she hits the iceberg. Let M be the number of ways for Lily to fire N missiles and guarantee that she hits the iceberg. Compute 100M+N

Answer 1

$\text{Let x and y sides of the lake}$

$\text{The minimum number of shots required is 4:}$

$\text{1.-this is a splitting of only sides x or y into pieces of length at most 2}$

$n = \text{integer}9/2 = 5$

$\text{2.-Or a pairwise partition of x and y into pieces of length at most 4}$

$n = \text{integer}9/4+\text{integer}9/4= 6$

$\text {We take into account that the number of shots is 1 less than the}$

$\text{number of parts for each side of the lake}$

$\text{For option 1. we have the following methods}$

$\text{(1,2,2,2,2);(2,1,2,2,2);(2,2,1,2,2);(2,2,2,1,2);(2,2,2,2,1);}$

$\text{That is, 5 * 2 =10 variants for x and side y}$

$\text{For option 2. we have the following possibilities of splitting into parts:}$

$\text{(1,4,4);(2,3,4);(2,4,3);(3,2,4);(3,4,2),(3,3,3)}$

$\text{(4,1,4);(4,2,3);(4,3,2);(4,4,1);}$

$\text{It is possible to choose 1 out of 10 for each side of the lake}$

$\text{The total number of options is} 10 * 10=100$

$M = 100+10 =110$

$N=4$

$100*M+N = 110*100+4 = 11004$

Answer:11004