Numbers on lottery ticket
Source: 2023 Grosman MO, P2
September 8, 2023
combinatoricsInfogrid
Problem Statement
A "Hishgad" lottery ticket contains the numbers to , arranged in some order in a table with rows and columns. It is known that the numbers in each row increase from left to right and the numbers in each column increase from top to bottom. An example for and :[asy]
size(3cm);
Label[][] numbers = {{"", "", "", ""}, {"", "", "", ""}, {"", "", "", ""}};
for (int i=0; i<5;++i) {
draw((i,0)--(i,3));
}
for (int i=0; i<4;++i) {
draw((0,i)--(4,i));
}
for (int i=0; i<4;++i){
for (int j=0; j<3;++j){
label(numbers[2-j], (i+0.5, j+0.5));
}}
[/asy]
When the ticket is bought the numbers are hidden, and one must "scratch" the ticket to reveal them. How many cells does it always suffice to reveal in order to determine the whole table with certainty?