A Fistful of Brain Teasers

Well, if it is not symmetrical b + c still equals 190 degrees.

Dozey posted:

Well, if it is not symmetrical b + c still equals 190 degrees.

agreed.

a = 55, b =125 using the half angle theorem c = 65 (360 - others)

Ah. I didn't know the half-angle theorem. Probably not taught in Scottish SMP maths. 

Guys,

I'm at work at the moment (lunch break). When I get home this evening, I will re-check my geometry book just to make absolutely certain that I haven't made an error in my presentation.

I'm pretty sure I haven't, but creating those little arcs to mark the angles is a right old pain in Powerpoint and I might have been distracted by it.

Fingers crossed !

Cheers, Don

Ian, Why did teacher not get back last night when you answered correctly. It does help to put your proof steps in. Like the half angle theorem solving a and b.  c then follows. I suppose the ordering of the questions is also a clue.

Teacher, your reputation is intact.

Ah, my last post crossed. It now looks like Phil has a solution that matches Ian's first solution.

Both match my solution.

I will still check this evening.

Don

Filipe posted:

Ian, Why did teacher not get back last night when you answered correctly. It does help to put your proof steps in. Like the half angle theorem solving a and b.  c then follows. I suppose the ordering of the questions is also a clue.

Teacher, your reputation is intact.

Ian put a condition into his response, vis "assuming BOD is a straight line"

I considered it worthwhile waiting for unconditional responses.............

Don, your integer equations (Diophanine equations) have stimulated me look at number theory on wiki. I am being forced to relearn matrices and Smith Normal Form. These branches of math are important in physics. There are online lectures for when the overly notational stuff which is not always consistently used across disciplines leaves you scratching your head.

Thanks

Phil

Dozey posted:

I think the potato one might be 7 - 1, 2, 4, 8, 16, 32, 64.

It can be done with fewer than 7

It can be done with potatoes, but could be messy with flour....................onless the flour was bagged !

Don Atkinson posted:

Find the marked angles a, b, and c. O is the centre of the circle.

This and the next few are designed to be very simple (after the trauma of the Road AF and the Notebook drama!).

Basically, he who can find his O-Level maths book the quickest, will be the winner !

a = 55°     Angle at centre = twice angle at circumference (both on arc AC)

b = 125°  Opposite angles of a cyclic quadrilateral ABCD

c = 65°    Angle sum of a quadrilateral OABC

Find the marked angles q and r.

TP is a tangent, touching the circle at P

RèeDon Atkinson posted:

Find the marked angles q and r.

TP is a tangent, touching the circle at P

Let the apex with angle q move around the circle. As it does the angle subtended theorem states the angle remains constant. Approaching the tangent point this is also true, therefore

q = 140

2r = 180 - q = 40

r = 20

Don Atkinson posted:

By intersecting chords theorem

3x**2= 48

x= 4

Filipe posted:

RèeDon Atkinson posted:

Find the marked angles q and r.

TP is a tangent, touching the circle at P

Let the apex with angle q move around the circle. As it does the angle subtended theorem states the angle remains constant. Approaching the tangent point this is also true, therefore

q = 140

2r = 180 - q = 40

r = 20

NiceLy set out Phil. Well done !

Don Atkinson posted:

It can be done with potatoes, but could be messy with flour....................onless the flour was bagged !

If you put it like that then he only needs to be able weigh 1lb accurately, but then the bagged potatoes become weights. You could say that the international kilogram works on the same idea!

Don Atkinson posted:

Dozey posted:

I think the potato one might be 7 - 1, 2, 4, 8, 16, 32, 64.

It can be done with fewer than 7

Well it can in theory be done with a single 1 lb weight if you have time on your hands. (I see Filipe got there first!)

OK, how about 1, 3, 9, 27, 81. That comes to 5.

Dozey posted:

OK, how about 1, 3, 9, 27, 81. That comes to 5.

That's more like it !

Well done Dozey

Don Atkinson posted:

Dozey posted:

OK, how about 1, 3, 9, 27, 81. That comes to 5.

That's more like it !

Well done Dozey

Yes, well done from me. I have not checked it but I understand the lateral thinking in the solution.

Phil

My first answer was for a weighing machine where you put all the weights on one side. The second is when you can put weights on opposite sides of a balance. I am sure there must be an analytic solution.

I'll come back to "Angles" later.......

......now for something completely different !

OK, it's only GPS.....ie the American version of GNSS (Global Navigation Satellite Systems)

GPS relies on accurate timing of radio transmissions from a constellation of NavStar satellites orbiting the earth. Sticking with the basic GPS, ie setting aside any of the augmentation systems such as WAAS :-

a) How many satellites in the minimum GPS constellation

b) How many are currently in the GPS constellation

c) How far above the Earth do the satellites orbit

d) How long does each satellite take to orbit the Earth

e) describe the satellite orbits

We'll come to the real brain teasers next, ie the all-important timing of radio transmissions and aspects of relativity. Meanwhile the old Google button should lead to most of the answers although some are more reliable and more accurate than others !

a. 24

b. 32, although 1 isn't working

c. 20,200 km

d. 11 hours 58 minutes (0.5 sidereal days)

e. they are all great circles, lying in 6 different planes, with 55 degree inclinations to the equator.