Arithmetic sequence square
Source:
March 17, 2006
Problem Statement
It is possible to place positive integers into the vacant twenty-one squares of the square shown below so that the numbers in each row and column form arithmetic sequences. Find the number that must occupy the vacant square marked by the asterisk (*).[asy]
int i;
for(i=1; i<5; i=i+1) {
draw((0,2*i)--(10,2*i));
draw((2*i,0)--(2*i,10));
}
string[] no={"0", "74", "103", "*", "186"};
pair[] yes={(1,1), (3,7), (5,3), (7,9), (9,5)};
for(i=0; i<5; i=i+1) {
label(no, yes);
}
draw(origin--(10,0)--(10,10)--(0,10)--cycle, linewidth(2));[/asy]