A Fistful of Brain Teasers

Innocent Bystander posted:

It suggested to me that the four triangles in this basic shape will always be equal regardlee of the sizes of the squares - so as on this occasion I’d used a spreadsheet to do the calcs, before closing I substituted three random numbers for the areas of the squares - and yes, indeed, the four triangles come out with the same areas.

 

It's true that all 4 triangle will have the same area. In the case of Farmer Watts, this was 9 acres. This provides a very satisfying solution to the overall acreage of 100.

But putting any three square fields together in the form of the initial triangle, then adding the three external triangles leads to four new fields, each with the same area.

I haven't been able to find a neat set of initial fields, other than the 26, 20 and 18 that gives such a satisfyingly precise overall acreage as 100. One or two close ones, but not 100.

Innocent Bystander posted:

I’ve scribbled the calcs, and done a diagram to illustrate - I’ll upload both when I’ve got time for the hassle of uploading the image 

I load my scribbles into Powerpoint and save them as JPEG.

I then load the JPEG into Flickr and transfer from Flickr to Naim using the <> tool in the top-right part of the Forum Post toolbar.

It's a right old pain !

Don Atkinson posted:
Innocent Bystander posted:

I’ve scribbled the calcs, and done a diagram to illustrate - I’ll upload both when I’ve got time for the hassle of uploading the image 

I load my scribbles into Powerpoint and save them as JPEG.

I then load the JPEG into Flickr and transfer from Flickr to Naim using the <> tool in the top-right part of the Forum Post toolbar.

It's a right old pain !

I use Imgur - doesn’t need an account, but still a hassle!

Farm Estate Spreadsheet JPEG

My spread sheet approach.      (a bit messy-looking, but it works !)

Clearly there are formulae in the cells with the red number "9"s    (and many of the other cells)

eg the first one (Heron's law) the cell contains "= (E10*H5*H6*H7)^0.5"        

in other words    √(6.906898 x 1.807879 x 2.434762 x 2.664257)

Lost count of how many attempts to load image so far, but I’ve spent far too long on it. So, holping this is OK by forum rules, here’s a link that should pull up the associated diagram:

https://m.imgur.com/a/LHDxk

NOtation below relates to this image, not any others.


a= SQRT 18
b= SQRT 26
c= SQRT 20


Internal triangle:
Cosine rule I had to look up, for angles of non-right-angled triangle with all sides known
Cos B = ( c^2 + a^2 – b^2 ) / 2ac
= (20 + 18 – 26) / 2 x sqrt 18 x sqrt 20
Angle A = 52.12 deg
Angle B = 56.31 deg
Angle C = 71.57 deg
Internal triangle height h
Sin C = h/a, so h = a x Sin C
Area = 0.5 x b x h
=9

External triangle:
Angle D = 360-90-90 –B = 123.69 deg
Angle E = 180- D = 56.31 deg
Height of external triangle (k)
Sin E = k/c, so k = c x Sin E
Area external triangle = 0.5 x base x height
= 0.5 x a x (c x Sin E)
=9

Similarly for other external triangles.

Innocent Bystander posted:
Don Atkinson posted:

Time flies ? You can’t ! They fly at such irregular intervals !

Well done! Maybe not as much of a challenge as I thought it might be - adults often get hung up on tempus fugit rather than insects. It is actually a question from an 11-plus exam in the  1950s

Ah ! Now, I did my 11-plus exams, there were two stages, in the late 1950s. But I don’t recall that particular question   

Mine wasn’t until mid 60s. I seem to recall (rather vaguely) there was an “intelligence test” with various puzzles, and maths and English. That one was from someone I knew in the late 1970s, some years older than me, who remembered it from his own 11-plus, but I don’t know exactly whixh year.

dave marshall posted:

Confused?

Surely not that straightforward? 

"Well, after that struggle with Farmer Watts and his 100 acre field system, I thought something a tad more "gentle" on these old brains might be in order......."  (that was my introduction to these last few teasers)

Not everything has to be confusing dave ! Somethings in life can be straightforward. And this, in itself, sometimes surprises us.

But as you can see, what one person considers straightforward, others don't.

Well done !

Innocent Bystander posted:
Don Atkinson posted:

D E S U JPEG

Another "language" one, especially since IB seems to prefer these to "maths" ones

The best I can do is T (NOT FUSED), but somehow I don’t think that was what was intended, HVe to look again when my brain is fresh.

Now if the S was an O...

I guess by now you have seen dave's contribution ?

But there might well be other solutions.

Hi NFG,

I am struggling slightly to grasp your teaser. This could just be me having a mental block !

I visualize two goats, each tethered such that Goat No.1 can eat a circle of grass radius 30m ie 2829 sqm. A second Goat can eat a circle of grass radius 40m ie 5029 sqm. These two circles overlap. The overlap being an area of grass common to both Goats. This area could vary from zero ( where the posts are 70m or more apart) up to 2829 sqm (where the posts are no more than 10m apart).

However, this area of grass, common to both goats, is specific in terms of square metres. For example, if it is 283 sqm (ie 10% of the smaller circle, then it is only 5.6% of the larger circle.

Can you appreciate my mental block, given the 10% wording of your teaser ?

Or, more likely, am I just being dumb !

Innocent Bystander posted:

The only explanation I could envisage is the larger circle being close enough to a boundary wall to restrict its area. But if that is the case, you could equally well have them both partially cut off by walls, with any number of answers.

Looks like you are having similar "mental blocks" ?

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