Brain Teasers are Back!

We’re you affected by the earthquake that damaged Morroco ? The fault line seems to run south west along the Atlas Mountains towards the Canary Islands.

I think there’s only on way to do it…halve N-S longitudinally, halve E-W longitudinally, then along the equator. As it were.

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Plus there’s the standard Edam wedges, slice longitudinally every 45 degrees.

I haven’t worked out the amount of cling film but I think it would be more than the shape made by @thebadyogi

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That’s exactly what I did. I described it as ’ halve it laterally ie along the equator. Then quarter the northern hemisphere and likewise the southern hemisphere’.

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It’s fairly straight forward to “visualize” which of the two solutions has the smaller surface area, and you don’t need to know the surface area of a sphere to work out this difference. Also, the difference can be easily described very clearly in a few words, without doing any real arithmetic.

But to work out the actual area of cling film (surface area) does need a bit of maths …

Ok, a bit more maths …

… that is precisely what Mrs D wanted … (but wasn’t what she requested !!)

Fortunately, it’s what I did … a bit like eight segments of an orange.

There has been no impact to holiday makers, as far as I can tell.

The Canary Islands are fairly close to Morocco, so there are quite a few Moroccans living here who no doubt will have been indirectly affected.

Firstly a bit of lateral thinking. @thebadyogi used three cuts and the standard wedges need four cuts, all of which are full circles through the middle.

I will leave the maths for others - I have just finished my second sangría and Mrs R now wants me to mix a gin and tonic as an aperitif. I am more pie-eyed than π r squared.

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Couldn’t visualise that at all but it’s so obvious now!!

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Intuitively it would seem that more cuts means more surface area of cheese vs wax so I agree with you. I am not a mathematician😂

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I’m sure that a few simple words would suffice, but …

Generally

I have a spherical cheese Radius R. Its surface area = 4πR²
If I cut this spherical cheese Radius R in half, I get two hemispheres
Each has a flat-disc-base of exposed cheese, area πR²
Each has a hemispherical wax surface, area 2πR²

Option 1

If I cut each hemisphere of cheese into 4 equal segments, as per an orange …
… I will get 8 segments. Each segment will have two half-discs of exposed cheese ie an area the same as a full, flat-disc-base = πR². Each segment will also have an area of wax equal to 1/8 the area of the sphere (or ¼ the area of a hemisphere) either way = ½πR²

Alternatively …

Option 2

If I place each hemisphere of cheese with the flat disc of exposed cheese, face-down, and then quarter each hemisphere with cuts N-S then E-W, I will get 8 pieces of cheese, each with 3 flat, quarter-discs of exposed cheese. Total area of exposed cheese on each of the eight pieces will be ¾πR². Each piece of cheese will also have an area of wax equal to 1/8 the area of the sphere (or ¼ the area of a hemisphere) either way = ½πR² (ie the same area as per the segment option)

Conclusion

Both options have the same area of waxed cheese.
Option 2 has a smaller surface area of exposed cheese than Option 1
Viz; ¾πR² v πR²

In other words, Option 2 has 3 x quarters of a flat disc of exposed cheese, whereas Option 1 has 4 x quarters of that flat disc of exposed cheese. They both have the same area, but different shape of wax surface. This should be easy enough to visualise without the maths

I wasn’t sure what you had intended, as ratios are dimensionless so can’t apply to areas vs volume. Maybe ratio of wax to exposed cheese perhaps?

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Hi Ravvie (and others who might be browsing …)

I understand your concern and uncertainty. You are. of course, quite correct that ‘ratios’ are dimensionless. That is why I added the words ’ … of surface area to volume …’ This part of the teaser was simply to allow Form Members who knew (or could look up) the surface area of a sphere to do a bit of maths

The main aim of the teaser was to visualise two ways of cutting the cheese. Then visualise that in both cases, the area of wax was the same (1/8 of the sphere). Then again visualise, that your way (eight orange segments) (#) exposed an unwaxed area = to one hemispherical disc, whilst ‘thebadyogi’ way (#) exposed an unwaxed area = to three-quarts of that hemispherical disc.

I hope that thebadyogi, yourself and one or two others enjoyed the teaser, despite it’s limitations.

(#) I appreciate that both yourself and thebadyogi had visualised each of the two possible shapes and the 1 : 3/4 ratio of unwaxed cheese.

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I was probably being a bit mathematically pedantic, reversion to type I suppose!

Yes it was a good teaser. Three of us in the R family tried very hard to visualise a third solution but with no success.

I had to ask Mrs R what the formula was for surface area of a sphere, though I had remembered 4/3 π R^3 for the volume. She just said “to get from volume to area, just differentiate it”. Silly me, I should have remembered that! Maybe that won’t mean much to Brain Teaser followers that haven’t done calculus but it leads to 4 π R^2.

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Yes. Most of us can remember π R^2 and 4/3 π R^3 but not many of us seem to recall 4π R^2 for the surface of a sphere.

My method is to recall that if you have a closed hemisphere, imagine the flat-disc-circular-base, area
πR^2 is elasticated. If inflated, or pushed inwards to fill the hemisphere, it has to stretch to TWICE its flat disc size 2πR^2 ! Double this gives the area of the sphere.

I think this factor of x2 helps to put Mercator’s and Lambert’s map making efforts into perspective.

When we look at a photograph of the Earth, we see a flat disc, area ‘E’. The actual surface area of that half of the Earth is ‘2E’. Any map-maker has his work cut out

Nice one Ravvie

… to which the answer is …

Well, just to tidy this one up …

For the Ravvie ‘segment’ solution the ratio of wax divided by exposed cheese is 1/2