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)
Ian G. posted:Don Atkinson posted:Any body going to tackle Ian’s neat puzzle searching for Pi ?
To be fair any credit belongs to Archimedes !
He doesn’t seem to post on this forum these days....... 
Don Atkinson posted:Ian G. posted:Don Atkinson posted:Any body going to tackle Ian’s neat puzzle searching for Pi ?
To be fair any credit belongs to Archimedes !
He doesn’t seem to post on this forum these days.......
I’m sure he’s still floating about somewhere.
Ian G. posted:Don Atkinson posted:Ian G. posted:Don Atkinson posted:Any body going to tackle Ian’s neat puzzle searching for Pi ?
To be fair any credit belongs to Archimedes !
He doesn’t seem to post on this forum these days.......
I’m sure he’s still floating about somewhere.
2.598 & 3.674
assuming I understood the question and assuming Pythagoras had done his bit before Archimedes. (Area of inner hex touching circle = 2.598r^2, outer is 3.674r^2)
having scribbled workings on phone screen I managed to delete when I went to copy for her, and this time of night not doing again, but this is part: area of inner hex calculates out to be
A=6x 0.866r x 0.5r
Don Atkinson posted:
1 big box (5 units)
2*2 creating the chevron pattern
2*4 + 1
1*3
2*2 in centre
1*3 in centre
total 18
Filipe posted:Don Atkinson posted:
1 big box (5 units)
2*2 creating the chevron pattern
2*4 + 1
1*3
2*2 in centre
1*3 in centre
total 18
Darn it, IB you are right
1 big box (5 units)
2 + 2 creating the chevron pattern
2*4 + 1 (1)
2 (2) top right and bottom left
2 (2) in centre
1 (3) in centre
total 19
Innocent Bystander posted:Ian G. posted:Don Atkinson posted:Ian G. posted:Don Atkinson posted:Any body going to tackle Ian’s neat puzzle searching for Pi ?
To be fair any credit belongs to Archimedes !
He doesn’t seem to post on this forum these days.......
I’m sure he’s still floating about somewhere.
2.598 & 3.674
assuming I understood the question and assuming Pythagoras had done his bit before Archimedes. (Area of inner hex touching circle = 2.598r^2, outer is 3.674r^2)
having scribbled workings on phone screen I managed to delete when I went to copy for her, and this time of night not doing again, but this is part: area of inner hex calculates out to be
A=6x 0.866r x 0.5r
That works, but there is a better (tighter) pair of bounds you can get from same figures.
Ian G. posted:Innocent Bystander posted:Ian G. posted:Don Atkinson posted:Ian G. posted:Don Atkinson posted:Any body going to tackle Ian’s neat puzzle searching for Pi ?
To be fair any credit belongs to Archimedes !
He doesn’t seem to post on this forum these days.......
I’m sure he’s still floating about somewhere.
2.598 & 3.674
assuming I understood the question and assuming Pythagoras had done his bit before Archimedes. (Area of inner hex touching circle = 2.598r^2, outer is 3.674r^2)
having scribbled workings on phone screen I managed to delete when I went to copy for her, and this time of night not doing again, but this is part: area of inner hex calculates out to be
A=6x 0.866r x 0.5r
That works, but there is a better (tighter) pair of bounds you can get from same figures.
2*sqrt(3) > pi > 3
The inner equilateral triangles have chord of 1 (Unit circle). The outer is obtained with similar triangles and Pythagoras. Larger polygons give more accuracy.
Phil
Ian G. posted:Innocent Bystander posted:Ian G. posted:Don Atkinson posted:Ian G. posted:Don Atkinson posted:Any body going to tackle Ian’s neat puzzle searching for Pi ?
To be fair any credit belongs to Archimedes !
He doesn’t seem to post on this forum these days.......
I’m sure he’s still floating about somewhere.
2.598 & 3.674
assuming I understood the question and assuming Pythagoras had done his bit before Archimedes. (Area of inner hex touching circle = 2.598r^2, outer is 3.674r^2)
having scribbled workings on phone screen I managed to delete when I went to copy for her, and this time of night not doing again, but this is part: area of inner hex calculates out to be
A=6x 0.866r x 0.5r
That works, but there is a better (tighter) pair of bounds you can get from same figures.
I can see that with the inner one it should be simple to calculate construct 6 equilateral triangles with a side of the hexagon as base and apex on the circumference and calculate their areas as a function of the radius, to combine with the hex area - but that is going beyond the brief (You can find upper and lower bounds on the value of π geometrically using a circle and a couple of hexagons - one for the lower bound and one for the upper bound. What are the values of these two bounds on π ? ) I'm struggling to see how to get further with just 2 hexagons and a circle.
Just to help any on-lookers, this and the next few slides might make the counting a bit easier..... !
Take care with this one ! People often fail to spot the blue and yellow squares
Again, take care, people often fail to spot the middle one and consequently the associated red dot !
These two don't usually cause problems.