A Fistful of Brain Teasers

Spreadsheet to follow tomorrow............(hopefully)

You can create your own spreadsheet to check mine. I did mine in "bits" eg R/2; R² etc so that I could remember what I was doing.

I have a feeling that Excel has a macro called "Goal-Seeker" or something. Should be able to find a "Zero" solution in a couple of seconds. I did mine by hand.

Remember, you need to multiply the Excel output by 100 to get the length of rope.

I though I'd better go back to a more straight forward Teaser for the start of a new year, especially since any hangovers might need an extra "hair of the dog" chaser if we put in Bam's Ladder or Adam's bowl of water.....

We were reminiscing, as you sometimes do, on New Years Eve.

Mrs D remarked that when our neighbours, John and Joan got married eighteen years ago, John was three times as old as Joan. Now he is only twice as old as she.

So how old was Joan on her wedding day ?

Another easy one to ease into the New Year...

How many minutes is it until six o’clock if fifty minutes ago it was four times as many minutes past three o’clock ?

Don Atkinson posted:

So how old was Joan on her wedding day ?

Hi Don,

Joan was 18 years old on her wedding day. John was three times as old back then, which makes him 54. Today they are 36 and 72 years old respectively.

Happy new year, by the way 

Hi Mulberry,

As always (ok, usually always) spot-on.

The age difference is improbable, but definitely not impossible and i'm sure there are a few living examples around !

Happy New Year to you too.

Cheers, Don

The crescent is formed by two circles. C is the centre of the larger circle. The width of the crescent between B and D is 9 inches, and between E and F 5 inches. The centres of the two circles lie on the line AB, so there is a N/S symmetry in the diagram above ie the 5" EF dimension is repeated between the two circles below C

What are the diameters of the two circles ?

Don Atkinson posted:

Another easy one to ease into the New Year...

How many minutes is it until six o’clock if fifty minutes ago it was four times as many minutes past three o’clock ?

 

I’ll take the other easy one as well and say it is (was?) 26 minutes until six o’clock, otherwise known as 17:34 Fifty minutes ago it was 16:44 or 104 minutes past three o’clock.

Mulberry posted:

Don Atkinson posted:

Another easy one to ease into the New Year...

How many minutes is it until six o’clock if fifty minutes ago it was four times as many minutes past three o’clock ?

 

I’ll take the other easy one as well and say it is (was?) 26 minutes until six o’clock, otherwise known as 17:34 Fifty minutes ago it was 16:44 or 104 minutes past three o’clock.

Yet another masterly response brings  a Teaser to its knees.

Well done Mulberry !

The "Crescent" isn't that  difficult, well, not like the goat in the circular field.......but it does depend on a little bit of geometry

If there is no solution or debate in progress over the weekend, I'll post a few hints.

Presumably the geometric principle is that a triangle drawn from the diameter of a circle will have a right angle at the circumference. Then I’m guessing it’s a case of using pythag to generate a series of simultaneous equations to solve for R and r.

Hi SJ,

Nice to hear from you again. I would try an easier approach......(well I would say that, wouldn't I )

Note the distance of Point "E" from the outer circle.

Roll the inner circle left, until it becomes concentric with the outer circle.

Now try looking at the geometry again to see if the diameter (or radius) of the inner circle can be calculated.

It can involve right angled triangles, but not necessarily using the principle that you were considering - which I'm glad you mentioned, because many will have forgotten that !

Cheers, Don

And a happy new year !

Don Atkinson posted:

The crescent is formed by two circles. C is the centre of the larger circle. The width of the crescent between B and D is 9 inches, and between E and F 5 inches. The centres of the two circles lie on the line AB, so there is a N/S symmetry in the diagram above ie the 5" EF dimension is repeated between the two circles below C

What are the diameters of the two circles ?

let

center of small = O

r = small rad

R=  big rad

2R-2r =9 or R-r =4.5 =OC;

r=R-4.5

also

EC= R-5

OC^2+EC^2 = r^2  i.e.

4.5^2+ (R-5)^2=r^2=(R-4.5)^2 ;

4.5^2 + R^2+25-10R = R^2+4.5^2-9R

R= 25, r=20.5 i.e. big circle dia=50, small = 41 ?

Hi Brilliant,

That is the neat solution. Well done, Top Man !

I'll post below, the diagram which I have, that I think outlines your solution. Others might find it helpful.

There are a couple of other ways of solving this problem which I shall describe later (because I haven't done the diagrams yet !)

Brilliant's solution in picture format

At least I hope this is the same as Brilliant's solution

(Note: "D = Diameter of Small circle" should read "d = diameter of small circle"..........apologies!)

Don Atkinson posted:

Hi Brilliant,

That is the neat solution. Well done, Top Man !

I'll post below, the diagram which I have, that I think outlines your solution. Others might find it helpful.

There are a couple of other ways of solving this problem which I shall describe later (because I haven't done the diagrams yet !)

Hi Don,

Thank you for that! It was -21C with windchill down to -38C outside, so I fancied this might be a friendlier challenge. Diagram is a beauty, looking forward to the other ones!

This is my second solution.

It's the one I hinted at for sjbabbey. It involves "sliding" the inner circle left by 4.5 inches and appreciating that HG = half an inch (= my comment  little bit of geometry).

The rest should be easily followed.

I do have another solution..................(same answer of course !)

Brilliant posted:

Hi Don,

Thank you for that! It was -21C with windchill down to -38C outside, so I fancied this might be a friendlier challenge. Diagram is a beauty, looking forward to the other ones!

Wise decision, Brilliant !

Mt daughter lives in Coldstream, Vernon BC and they spent the weekend skiing up at Silver Star Friday Saturday then snow clearing Sunday (they manage a number of holiday rental properties).

We and daughter No 3 are due out in a couple of weeks and the the snow at Silver Star looks good.  The other set of "in-laws" are in Canmore so we'll also face the TransCanada drive (actually, for winky's benefit, we look forward to it) and enjoy some fun up at Sunshine.

You need to have a purpose to face -38C

Don Atkinson posted:

This is my second solution.

It's the one I hinted at for sjbabbey. It involves "sliding" the inner circle left by 4.5 inches and appreciating that HG = half an inch (= my comment  little bit of geometry).

The rest should be easily followed.

I do have another solution..................(same answer of course !)

Anybody remember the "Intersecting Chords Theorem" ?

No need to "prove" it, just "use" it.

This should help a few people to solve the Two Circle - Crescent Teaser on the previous page and above

......hopefully !

BTW it's easy enough to prove the Intersecting Chords Theorem (or you can just look it up on the internet !)

The "Intersecting Chords" are pretty unique.

a = b

and the two chords are at right angles

But the theorem still holds true.

Of course, you still need to appreciate how to show HG = 0.5"

Domestic HiFi row ! (row - as in quarrel, not row as in paddle a boat)

A woman is sitting in a boat in a small reservoir. In the boat with her is a hemp sack containing the dead body of her audiophile husband. She had become frustrated with her obsessive partner mainly due to the proliferation of ugly black boxes and metalwork that had taken over her living room. Not to mention the brick dust and ill fitting carpets where he installed the make-shift mains spur; nor to mention the devastation of her beautiful flower beds where he had insisted on driving a dozen copper earthing pipes into the ground.

Anyhow, I digress. She has weighed the sack down with six 135s and the twisted remains of his Mana stands and sewn it tightly with NACA5 (ensuring correct direction in the weave) to ensure he will sink without trace. Whilst securing the parcel she glances over to the reservoir wall and notices a water level marker. With a major struggle she manages to heave the sack of hubby and hifi over board. After a while she settles back in the boat and smokes a well-earned cigarette. The reservoir wall catches her eye again; she notices the water level marker.

Has the water risen, fallen or remained the same against its original level on the water level marker?

PS. no water has entered or left the reservoir etc

.....and the ciggarette smoking is insignificant !

I had thought that using the "modern" exam multi-choice technique, that answers, both right and wrong, would have flooded in (there was the makings of a pun in there somewhere !)

so i'll make the choice easier a), b) or c)

• a) the water level rises

• b) the water level remains constant

• c) the water level falls

I'll leave the need for an explanation till later.

Hi Don,

the water level goes down, which is not what I thought at first. Both my wife and I liked this one, so I had to try it out in our kitchen (wife and Hi-Fi are completely unharmed). Not sure why this is the case at the moment, although I have an idea which I want to think over.