*The questions are divided into four phases, based loosely on those found in George Pólya's 1945 book "How to Solve It".*Understanding the problem Finally, don't forget that STEP questions are designed to take at least 30-45 minutes to solve, and to start with they will take you longer than that.All together, we know the sofa constant has to be between 2.2195 and 2.8284." The Collatz conjecture is one of the most famous unsolved mathematical problems, because it's so simple, you can explain it to a primary-school-aged kid, and they'll probably be intrigued enough to try and find the answer for themselves. A common prime factor means that each of the numbers needs to be divisible by the same prime number.

It sounds obvious that the answer would be yes, after all, 3 1 = 4, 5 1 = 6 and so on.

We know there are seven days in the week, so: d e = 7 And she trains 27 hours in a week, with d 5 hour days and e 3 hour days: 5d 3e = 27 We are being asked for how many days she trains for 5 hours: d Solve: The number of "5 hour" days is 3 Check: She trains for 5 hours on 3 days a week, so she must train for 3 hours a day on the other 4 days of the week.

3 × 5 hours = 15 hours, plus 4 × 3 hours = 12 hours gives a total of 27 hours So Joel’s normal rate of pay is per hour Check Joel’s normal rate of pay is per hour, so his overtime rate is 1¼ × per hour = per hour.

Rather than giving up and just buying a beanbag, at this point, mathematicians want to know: what's the largest sofa you could possible fit around a 90 degree corner, regardless of shape, without it bending?

(Although they're looking at the whole thing from a two-dimensional perspective.)Thompson explains: "The largest area that can fit around a corner is called - I kid you not - the sofa constant. Eventually, if you keep going, you'll eventually end up at 1 every single time (try it for yourself, we'll wait). But the problem is that even though mathematicians have shown this is the case with millions of numbers, they haven't found any numbers out there that won't stick to the rules."It's possible that there's some really big number that goes to infinity instead, or maybe a number that gets stuck in a loop and never reaches 1," explains Thompson.

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