A Fistful of Brain Teasers
Posted by: Don Atkinson on 13 November 2017
A Fistful of Brain Teasers
For those who are either non-British, or under the age of 65………. The UK used to have a brilliant system of currency referred to as “Pounds, Shillings and Pence”. Simplified to £ ״ s ״ d. No! Don’t ask me why the “Pence” symbol is a “d”, just learn it and remember it !
A £ comprised 20 Shillings and a Shilling comprised 12 Pence. Thus a £ comprised 240 Pence. I reckon that both Microsoft and Apple would have difficulty with these numbers in their spreadsheets, more so if we included Guineas, Crowns, Half-Crowns and Florins. However, I digress..............
The purpose of the explanation is to assist with the first two or three teasers that follow. So just to ensure a reasonable comprehension has been grasped…. ….. if each of three children has £3 − 7s − 9d, then collectively they have £10 − 3s − 3d Got the idea ? Good ! Just try 5 children, two each with £4 − 15s − 8d and three each with £3 − 3s − 4d. How much do they have between them ? (this isn’t the first brain teaser, just the basic introduction with some “homework”, the Teasers follow)
Hopefully it's readable ? if not, it's just to find the missing numbers "X" when 124X is divided by X3 giving 2X precisely.
Note : "X" simply replaces the number that is missing. Not all the "X"s are necessarily the same !
Hopefully a bit bigger and clearer than the previous version, but the numbers are still the same.
And yes, I know it's easy 
Now for something a little more challenging 
Four Circles
Its (pretty) obvious that the maximum number of different sized circles (excluding infinitely large circles that have become straight lines!) that can be made to 'just touch' one another is four. (each and every circle just touches the other three).
In the attached diagram above, the three circles (Yellow), (Green) and (Blue) have radii of 1, 2 and 3 respectively.
What are the radii of the two 'fourth' circles (shown in red in the diagram below), that can be made to just touch each of Yellow, Green and Blue. (note the two 'fourth' circles will not touch each other, ‘cos that would break the maximum rule!!)
Oh! BTW, there is a standard way and an elegant way to solve this little problem.
Any more editing of this question Don, and there are going to be stiff letters to the Telegraph demanding that the head of the examining board is sacked 
C.
Christopher_M posted:Very confused by question.
Yellow r = 1, green r = 2, Blue r = 3
Ah, I think you may have just changed the lower diagram.
So you want the radius of the red circle and what I'm going to call the white circle? Jeez!
"Yellow r = 1, Green r = 2, Blue r = 3"........this part of the question Chris, you've got absolutely correct - well done !!!
But IB had complained once too often that the questions were too easy (easy for some ? perhaps !)
So on this occasion I added the little solid red circle and the big "empty" red circle, just to make this question a little bit more of a challenge than simply associating "1" with "yellow", "2" with Green etc....
And in doing so, I made a mistake, which I considered I should rectify to avoid confusion
And yes ! you have correctly identified the required answer(s) ie the radius of the red circle and what you are going to call the white circle?
I hope this is all crystal clear, ?
Jeez ! the effort I have to put in these days just to please some people.....
enjoy, (as ken c would say !) hopefully.........
I hope the "Head of the Examining Board" has avoided a charge of "Inappropriate behaviour" ?
Christopher_M posted:*behaviour* ;-)
Who is confusing who ?
Adam Meredith posted:Don Atkinson posted:Christopher_M posted:*behaviour* ;-)
Who is confusing who ?
Is the space before the '?' for the missing 'm'?
Pedant that.
I would have said “ idiosyncratic” if I thought I could spell “idiosyncratic” and was sure it was the right word to use
Don Atkinson posted:
Hopefully a bit bigger and clearer than the previous version, but the numbers are still the same.
And yes, I know it's easy
Nobody looks back a page when reading the Forum, so I thought i'd bring this and the next one forward...
Don Atkinson posted:
Now for something a little more challenging
Four Circles
Its (pretty) obvious that the maximum number of different sized circles (excluding infinitely large circles that have become straight lines!) that can be made to 'just touch' one another is four. (each and every circle just touches the other three).
In the attached diagram above, the three circles (Yellow), (Green) and (Blue) have radii of 1, 2 and 3 respectively.What are the radii of the two 'fourth' circles (shown in red in the diagram below), that can be made to just touch each of Yellow, Green and Blue. (note the two 'fourth' circles will not touch each other, ‘cos that would break the maximum rule!!)
Oh! BTW, there is a standard way and an elegant way to solve this little problem.
As I said above, No need to look back !!
Christopher_M posted:Don Atkinson posted:Using 1/r to represent "Curvature" is something I find most people have forgotten about !Always assuming they knew it in the first place :-)
eeK !!!.......Chris, what sort of school did you go to ! .......(probably had to settle for Harrow rather than Eaton ?) but at least NOW you know
And the circles with radii 1, 2 and 3 work out in a very satisfying way when using what I think is a rather elegant formula. Keep everything in vulgar fractions (none of this modern decimalisation) and they virtually cancel themselves out. All very neat.