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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.

In this podcast author Coralie Colmez shares insights into her novel

*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!

E=5 because multiples of 5 end in 5 or 0

Alternate digits must be even, so the rest have to be odd.

Digits C and D have to go "odd,even", and make a number which is a multiple of 4. So D has to be 2 or 6.

Ditto position H- must be 2 or 6.

The only even numbers left are 4 and 8, and these must go in B and F.

Looking at the first three digits, whose digital root must be 3,6 or 9, there are 9 options for filling these given the conditions we've already worked out.

We tried each of these in turn and worked out the digital root up to F. This also has to be 3,6 or 9 to make it divisible by 6. So you can work out in each case whether D is 2 or 6.

From this we can see what H is as one of its options has been used.

Only two digits remain- we test whether either makes a multiple of 7 when put in position G.

Finding that one of these works, check that the first 8 digits are divisible by 8.

After this only one answer remains:

381654729

(Also, has anyone noticed the patterns this and other suggestions make on the calculator buttons? They are symmetrical or generally interesting.)

If you followed that, I'm impressed. :)