The Phone Keypad Number Puzzle
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!
Jason Marshall, PhD
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The Phone Keypad Number Puzzle
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, 7×3=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.
Adding Rows
Any number that has a remainder of 1 added to any number that has a remainder of 2 must always give some number with a remainder of 3.
Now it’s time to get back to our original question: Why are all the sums of the horizontal rows, vertical columns, and diagonals always evenly divisible by 3?
To see, let’s think carefully about the numbers in the second row: 4, 5, and 6. When we add these three numbers up and divide the result by 3, we’re carrying out the arithmetic problem (4 + 5 + 6) / 3. But the distributive property says that we can distribute the division, and instead think about this as the problem 4/3 + 5/3 + 6/3. We know that 4/3 = 1 remainder 1, 5/3 = 1 remainder 2, and 6/3 = 2 remainder 0.
The sum of all this is equal to 4 remainder 3. But a remainder 3 when we’re dividing by 3 is the same as adding another whole number – which means 4 remainder 3 is actually equal to 4+1=5.
While all of this work was tons of fun, the truth is that we didn’t need to do it if our goal was only to show that the number is evenly divisble by 3. Because any number that has a remainder of 1 added to any number that has a remainder of 2 must always give some number with a remainder of 3 (which is really no remainder at all when we’re talking about division by 3.)
This is exactly the case for each of the three rows—which is exactly why the sums of all three rows are always divisible by 3. Make sense?
Adding Columns
How about the columns? Well, the numbers in the leftmost column all have a remainder of 1 when divided by 3, the numbers in the middle column all have a remainder of 2, and the numbers in the right column are all divisible by 3 from the get-go (which, according to the distributive property, means that their sum must be, too.)
So adding the three numbers in the leftmost column means you’re adding three numbers with remainders of 1 when divided by 3. In other words, the sum of these three numbers must have a remainder of 3 when divided by 3—or no remainder at all.
The numbers in the second column give you a remainder of 6 (since all three have remainder 2.) Since a remainder of 6 when dividing by 3 is really the same as adding a total of 2, the sum of the numbers in the second column must also give no remainder when dividing by 3.
The general rule is that whenever the division by 3 remainders of the trio of numbers you’re adding on the keypad add up to a multiple of 3 (and thus aren’t actually a remainder at all), their sum will be divisible by 3. And, as you can check, it turns out that this is true for all horizontal rows, all vertical columns, and both diagonals.
Is it true for any other trios of numbers on the keypad? I’ll leave it to you to puzzle that one out for a bit…
Wrap Up
Okay, that’s all the math we have time for today!
For more fun with math, please check out my book, The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of The Math Dude on Facebook, where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Phone keypad image from Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.