Question

It is possible to place positive integers into the vacant twenty-one squares of the $5\times5$ 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 $(*).$

Answer

Answer: 142

Concepts:arithmetic sequence pattern recognition system of equations

Difficulty rating: 2030

Small Hint:

Index the rows and columns from $0$ through $4$

Big Hint:

A grid whose rows and columns are arithmetic has the form $A+Bi+Cj+Dij$

Solution:

Index rows and columns from $0$ to $4.$ The row and column conditions give the general entry $A+Bi+Cj+Dij.$ The four shown values yield $\begin{aligned}A+4B&=0,\\A+B+C+D&=74,\\A+2B+4C+8D&=186,\\A+3B+2C+6D&=103.\end{aligned}$ Solving gives $A=52,$ $B=-13,$ $C=30,$ and $D=5.$ The asterisk is at $(i,j)=(0,3),$ so its value is $A+3C=52+90=142.$

1988 AIME Problem 6

Answer: 142

Difficulty rating: 2030

Small Hint:

Big Hint:

Solution:

Problem 6 in Other Years