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How is frequency related to pitch? Hear the music we love emerge from pure mathematical beats.

We talk to Stuart Johnston who uses mathematics to find out how noise pollution in the oceans impacts whales.

Generating electricity without the use of fossil fuels is not just an engineering and industrial challenge, it is also a huge mathematical challenge.

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*The irrational diary of Clara Valentine*.We talk to early career mathematicians who spent some of their summer holiday solving problems posed by industry — such as how to blend a perfect smoothie!

Let's omit the first stage of Kaprekar's algorithm and just look at what happens when the only rearrangement is reversing the digits in three of the numbers he looks at and doing a repeated operation.

Find the absolute difference between a number and its reverse, and then the difference between that and its reverse, and so on.

2005, 2997, 4995, 999

1789, 8082, 5274, 549, 396, 297, 495, 99

6174, 1458, 7083, 3276, 3447, 3996, 2997, 4995, 999

The absolute difference between the reverses of these repnines gives us our kernel, which is zero. That's true of all palindromes of course, though the operation in the algorithm producing a palindrome won't necessarily lead to a kernel. For example, consider summing them instead

2005, 7007

1789, 11660, 18271, 35552, 61105, 111221, 233332

6174, 10890, 20691, 40293, 79497

Lastly standard number line subtraction:

2005 ... (17 steps!) ... -8939779398

1789, -8082, -10890, -20691, -40293, -79497

6174, 1458, -7083, -10890, -20691, -40293, -79497 (A negative number can be regarded as a palindrome, eg -121 = 0-121-0)

Look at 6174 still behaving mysteriously, what's more in cahoots with that other notorious number 1089.

2005, 7007