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14:06 UK time, Friday, 23 December 2011

Is there any connection between Skye family's Christmas lights can be seen for miles and this story, Electricity failure affects Western Isles and Skye?
John, Scalloway, Scotland

Re the bonus question in your year-end quiz: after pictures 1 and 2, I just knew it had to be Apple, but you threw me with 3 & 4. Did Oliver Reed once have a scene with Glenda Jackson involving misuse of a fruit; and who is the woman in No.4 speaking about her husband's lovemaking prowess?
Michael Weir, St Albans, Herts
Monitor note: Not quite - the answer has now been revealed...

Oh how I wish I'd listened to Ralph and Warren (Wednesday letters) instead of Prof David Haslam ('Tell loved ones they are overweight this Christmas')! My mother-in-law just arrived for Christmas and gently passed on the advice to my wife and her mother. Still, on the bright side, at least it's warm for the time of year in Paris.
Jimlad, Cardboard Box, Champs Elysee, Paris

Roger (Thursday letters), one answer is that Pythagoras' Theorem is a special example of the cosine rule (look it up) which allows you to find the length of a side of a triangle provided you know the other two sides and the angle between them (try drawing two sides of a triangle with a certain angle between them: there's then only one length the missing side can be). When the angle you're using is 90 degrees (or a right angle, as in Pythagoras) then because cosine 90 is 0, the equation you use simplifies to Pythagoras' Theorem. It's a special case that looks too simple to be true.
I doubt that has made it any clearer so my second answer is "because".
Julia, Birmingham, UK

Roger, Danny Kaye did it in song!

"The square of the hypotenuse of a right triangle
Is equal to the sum of the squares of two adjacent sides.
You'd not tolerate lettin' your participle dangle,
So please effect the self-same respect for your geometric slides."

Susan Thomas, Brisbane, Australia

Roger, think of crazy paving. Take a right angled triangular slab and place three square slabs around it so that each square has sides equal to the side of the triangle it sits next to. The area of the two smaller slabs will be equal to the largest slab (which abuts the hypotenuse). Apologies, I may have been too serious for this letters page, I'll get my Santa hat...
Dave, Cambridge

@ Roger Perry - Friday's letters. If you think of it as literal squares on each side of the triangle it seems to make sense for me and gets you away from the mathematical equations. This website provides a nifty animated proof http://www.mathopenref.com/pythagorasproof.html
Karl, Isle of Man

First of all a very Merry Christmas to the BBC Magazine - Paper Monitor, Caption Comp et al - irrespective of gender or sentience. Second, today's random stat does not take account of quantum physics. Santa can move through every house and deliver presents in a fraction of a second which solves all those nasty physics problems like G-force and mass of presents. You see that's not fat under his uniform it's a quantum transport device. The sleigh and reindeer are merely decoys...
John Airey, Peterborough, UK

Angus (Thursday letters), who HATES Christmas, you wouldn't happen to be of Highland origin? Christmas was a non-event in the Highlands and Islands until relatively recently. I'm guessing you love New Year...
Buzz, London

Angus, oh! shame on you. I for one shall be awake at 4am to see if HE has been yet.
Graham (age 57 next week), Hayle, Cornwall

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