You know how your phone’s keypad is arranged into three rows and three columns of numbers - but did you know there’s a pretty cool math puzzle hidden in there, too? Keep on reading The Math Dude to find out how to find it, and solve it!
I recently received an email from math fan Rodney saying:
"One day, while waiting on hold for my TV provider, I noticed that if I add the numbers on the keypad of my phone in a straight line vertically, horizontally, or diagonally, the resulting sum is always divisible by 3. For example, vertically 1 + 4 + 7 = 12, and 12 / 3 = 4. I’m sure there is a simple explanation, but it’s taken me longer than the hold time to figure it out."
So, as Rodney asks, why does this happen? That’s exactly the puzzle we’ll be solving today.
The Phone (Or Calculator) Keypad
Pretty much every phone, calculator, and (at least older) computer keyboard contains a three column high by three column wide number pad.
On a phone, the top row contains 1-2-3, the middle row contains 4-5-6, and the bottom row contains 7-8-9. For some reason, calculators are always built the other way around, with the 1-2-3 on the bottom, the 4-5-6, in the middle, and the 7-8-9 on top. But no matter which way the keypad is constructed, the pattern that math fan Rodney found is present - if you look hard enough to find it.
That is, if you add the three numbers in any of the horizontal rows (which give sums of 1+2+3=6, 4+5+6=15, and 7+8+9=24), or the three numbers in any of the vertical columns (which give sums of 1+4+7=12, 2+5+8=15, and 3+6+9=18), or the three numbers along either of the two diagonals (which both give sums of 1+5+9=15 and 3+5+7=15), the resulting number will always be evenly divisible by 3.
Indeed, 6, 15, and 24 (the sums of the three horizontal rows), 12, 15, and 18 (the sums of the three vertical columns), and both 15s from the diagonals are all evenly divisible by 3.
Hmm…that's kind of curious, right? What’s up with that?
Division and Remainders
There are no doubt a number of ways to think about "what's up with that?," and to solve this puzzle, but today I’d like to focus on thinking about it in terms of division and remainders. As a reminder, whenever you divide one number by another—say 22/7—you can express the answer as a decimal. In this case, it'd be 3.142857…something…something…something.
Or, if you're not feeling up to doing all of that decimal division, you can write the answer as an integer with a remainder part. In this case, we can evenly divide 7 into 22 a total of 3 times…with 1 left over. So 22/7 is equal to “3 with a remainder of 1” - or put a bit more succinctly, “3 remainder 1.”
In other words, 7x3=21, but since we're trying to figure out how many times 7 goes into 22 (not 21), we're 1 shy (hence “remainder 1.”) All of which means that 22/7 is equal to "3 remainder 1" or, entirely equivalently, 3 and 1/7.
Remainders on the Phone Keypad
Now let's think about what happens when we divide the various numbers on the phone keypad by 3.
Let's start with the 1 in the upper-left corner: 1 can be evenly divided by 3 a grand total of 0 times, with a remainder of 1. How about the 4 immediately below the 1? Well, 4 divided by 3 is equal to 1 with a remainder of 1. And the 7 immediately below that?
Doing the math, we find that 7 divided by 3 is equal to 2 with—once again—a remainder of 1. Which means that all of the numbers in the first column—1, 4, and 7—are all divisible by 3 with a remainder of 1.
How about the numbers in the middle column? A quick bit of arithmetic will show you that 2 divided by 3 is equal to 0 remainder 2, 5 divided by 3 is equal to 1 remainder 2, and 8 divided by 3 is equal to 2 remainder 2. Again, the remainders of all the numbers in this column are the same - they're all equal to 2.
How about the numbers in the third column? Well, this one is pretty easy—3, 6, and 9 are all evenly divisible by 3, with a remainder of 0.