17 Hard Math Problems That’ll Make Your Head Spin

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Meghan Jones RD.COM Knowledge Psychology Updated: Nov. 05, 2021
These brain-teasing hard math problems are tricky, but they'll give your brain a workout—and they're really rewarding when you figure one out!
MIND STRETCHERS
Time to test your brain!
These hard math problems aren’t straightforward arithmetic. They’ll challenge you to look at the “problems” a different way and test your logic and problem-solving skills while you’re solving. And if math isn’t your strong suit, take heart—most of these hard math problems just use very simple numbers with only basic operations—addition, subtraction, multiplication, and division. What makes them a challenge is often examining the problem to find out the “trick,” or the pattern and the way the numbers relate to one another.
Others require a “fill-in-the-blank” technique that requires you to use trial and error and work backward. Time to put your brainpower to work! Maybe brush up on these easy math tricks you’ll wish you’d always known before you get started.
Track and field
If each of these runners travels the indicated number of spaces in the same amount of time, at which numbered spot will all of the runners be next to one another?
Answer: Space 19
At first, it seems like this will take forever to figure out—but, in actuality, the runners will all be lined up after only six “moves”! Make a chart indicating where each runner will be after each “move.” To figure that out, all you really need to do is add! Just add the number of spaces each runner advances—three, two, one, and five—to the number they’re already at, going back to Space 1 after 30. Sure enough, they’ll all reach Space 19 at the same time. In the mood for something harder? Keep going—or try these difficult brain teasers that will leave you stumped.
Ones and zeros
Below, you’ll see the binary notation for the decimal numbers 1 through 7. Following the pattern, what is the binary notation for the decimal number 12? Once you’ve solved this, brush up on some number facts you never knew before.
Answer: 1100
If you went to five digits for this one, you did it wrong (and that’s OK; that’s what I did on my first attempt)! What makes this difficult, and different from most hard math problems, is that it’s so unlike how we’re used to looking at numbers, especially in a sequence. Basically, you’re moving through all of the numbers you can make with just ones and zeros, starting with the least (0, represented as four 0s) and getting greater.
But you’ll always have four digits, so until you have a four-digit number (1000), the digit(s) to the left of the highest place will be a zero. So the question gives you up through 111 (written as 0111 to be four digits), and since it’s representing 7 and you’re trying to reach 12, just go through the next biggest numbers that use only 1s and 0s. So 8 will be 1000, 9 will be 1001, 10 will be 1010, 11 will be 1011, and 12 will be 1100.
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