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


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’.

I think I prefer your description. Well done thebadyogi !!

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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😂


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


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)


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 :sunglasses:

I follow your logic but your original question asked:

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 :sunglasses:

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 :sunglasses:

Nice one Ravvie

… to which the answer is … :sunglasses:

Well, just to tidy this one up …

For the Ravvie ‘segment’ solution the ratio of wax divided by exposed cheese is 1/2
For thebadyogi ‘Quadrant’ solution that ratio is 2/3

The reciprocals ie ratio of exposed cheese divide by wax, is 2 for the ‘Segment’ option
And for the ‘Quadrant’ it is 1 and 1/2

… but I think you knew all that !!

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And no πR^2 needed in the calculations!

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