- The Mind Your Decisions YouTube channel offers up a thorny math problem designed for 9-year-olds (!).
- Our solution involves narrowing down the possibilities using the logic of puzzle solving.
- The divisibility rule for 7 is a real mind-bender.
➗ You love challenging math problems. So do we. Let's solve them together.
It’s a letter problem, meaning each of the letters in the equation corresponds to a single digit.
Talwalker says teachers in Taiwan gave this problem to 9-year-olds, who understandably couldn't solve it—and neither could the teachers. So let's break it down:
Before we begin...
I can only walk you through how my mind puzzled out the math here. If you have a different idea or something more algorithmic, let us know in the comments. But what stood out to me is that E is the last value on the first line and in the solution. There aren’t a lot of digits you multiply by that result in themselves again, which meant it’s almost definitely a 5. Let’s substitute that and see what’s next. If it doesn't work, we can try something else.
This looks good to me. Now, we have another logical hook to unlock the next step. 555555 has a set number of factors, and we only have to test if it’s divisible by any of the positive integers less than 10. Obviously it’s divisible by 5, but we know A can't also be 5—all the “lettered” digits are different. It’s divisible by 3, but not 9.
There's no five-digit number we can multiply by 3 to get a six-digit number starting with 5. It’s not divisible by any even numbers. The highest value that A could have is 7. Is 555555 divisible by 7? Contextually, I feel it must be.
Some of the division rules for small integers are really easy. The one for 7, however, is not easy. Honestly, in this case, I’d just actually divide the number by 7, which is what I did in reality. But let’s also walk through the division rule for 7, because it’s interesting in itself, speaking of algorithmic thinking.
Here’s that rule, explained by a site called Byju’s:
So let’s process 555555 using the rule, iterating from our original number of 555555 all the way down to, hopefully, a two-digit number that's divisible by 7.
When we get down to a result of 42, we know the number 555555 is divisible by 7. Now we can just do the division and get the result: 79365.
When we multiply this through to check our work, it’s the right answer.
Talwalker solves the problem in a totally different way. Watch his great video and see that full solution, because he’s really doing the math in a comprehensive way:
My way, informed by decades of puzzle solving, involves narrowing the possibilities using logic and then checking my educated guess. Maybe you did something totally different. Sound off in the comments below. ⬇
Caroline Delbert is a writer, avid reader, and contributing editor at Pop Mech. She's also an enthusiast of just about everything. Her favorite topics include nuclear energy, cosmology, math of everyday things, and the philosophy of it all.