# 4×4 Grid and Extensions

Ben Vitale’s Fun with Num3ers ‘blog is a prolific source of all sorts of interesting number patterns.  He just posted a great problem that would be appropriate for students from elementary school through algebra.  Here it is:

Any students who understand nothing more two-digit addition could enjoy the magic that comes from getting the same answer every time.  Older students who are beginning to understand something about variables can handle the generalized question Ben asks.  Depending how one approaches the proof, a student might discover that this problem generalizes even a bit further than Ben suggests in his initial post.

Don’t read any further if you want to solve this problem on your own.

PROOF:  Let the number in the upper left of the grid be a.  One way to tackle this proof is to write the grid elements with the upper left number in parentheses, values added to that number along a row placed inside the parentheses, and values added to that number down a column placed outside the parentheses.  The revised grid looks like this:

Following the rules of selecting a number and then crossing out any other entries in that numbers row and column, every sum of four numbers selected this way will contain exactly one element from every row and every column making the overall sum contain an (a) from column 1, an (a + 1) from column 2, an (a + 2) from column 3, and an (a + 3) from column 4.  Also, every set of four numbers will have outside the parentheses nothing from row 1, a “+4” from row 2, a “+8” from row 3, and a “+12” from row 4.  That means the numbers you add for this sum will be some arrangement of $(a)+(a+1)+(a+2)+(a+3)+4+8+12=(4a+6)+24$.  Because $a=1$ for the given problem, the magic sum for this problem is 34. That solves an arithmetic problem.

EXTENSION 1:  Now think a bit more mathematically.  Notice that all my proof requires is that the upper left number be (a).  That means any consecutive integer run starting at any integer a in the upper left corner of a 4×4 grid would produce a constant sum of $4a+30$.  Encourage your mathematical explorers to start with or include all types of integers, including zero; include negative numbers if they’re ready for that.

EXTENSION 2:  How many different ways are there to pick numbers from a 4×4 grid in this manner, no matter what value (a) you place in the upper left corner?

EXTENSION 3:  Pushing just a little further, can you prove why any square grid of any size filled with any consecutive elements of any arithmetic sequence produces a constant sum?