Brain Teasers are Back!

It can be solved on a sheet of paper with a pencil, ruler, protractor and a (circle-drawing) compass.

Or a bit of trig.

Solution.

Draw the horizontal line labelled 270/GS. The length isn’t important but it needs to be about 200mm long. This represents the Desired Track.

At a convenient position, draw the line labelled 225/30 representing the wind speed and direction, making an angle of 45 deg (270 – 225) with the line 270/GS. I have labelled the intersection as X simply for reference.

Mark the position O on 225/30 such that OX = 30 units of length. I used 30 mm.

With a compass centred at O, draw an arc 120 units (mm in my case) radius to cut 270/GS at Y. This arc represents the True Airspeed using the same scale as the Wind Speed

You now have a triangle of velocities OXY

Wind Direction and Speed is represented by OX

Desired Track and Ground Speed by YX

Heading and True Airspeed by YO

A protractor centered at Y reveals required Heading 260 deg

Drift is therefore 10 deg Right

XY is 97 mm representing Ground Speed 97 kts

Trigonometry will deliver (unnecessarily) accurate angles and speeds.

The Octagon No 1

There are three empty rectangles.

The symbols +, - , and x (Plus, Minus and Multiply) are to be placed, one in each of the empty rectangles to make a sum that equals the number in the central rectangle, ie 76.

Each symbol must be used, and the calculation is made in the direction of the arrows, ie clockwise.

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-, +, X

17 - 19 = -2 + 40 = 38 x 2 = 76

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Nicely done Seakayaker. Nicely done.

Time in a Strange Land

The other night I had a dream. I had arrived in a strange land, but had a very helpful guide. When I awoke, I recalled the dream as follows -

As we approached the town hall, I noticed the clock on the outer wall of the tower. I commented that the minute hand was superimposed precisely over the hour hand and that since both lay between the 4 and the 5, I could tell him the time and did so !

Ah, exclaimed my guide, it appears that you are unaware that in this land of ours, whilst the hour hand rotates “clockwise” as in your land, the minute hand rotates in the opposite direction. Apart from this improvement, our clocks and yours are precisely the same !

What time was it ?

4:37?

Well Dozey, I can’t really argue with 4:37, well… I can ! so i’m only going to give you 8/10. at the moment. It is clearly correct to the nearest minute.

I rather suspect that you have a more precise answer written down. So 9/10 when I see the (rather) vulgar fraction, (or even a decimal to two places), and 10/10 when I see some sort of calculation or summary !

Nonetheless, well done !

Anyway, University Challenge is on in a moment. I used to give myself a pat-on-the-back every time I got a question right. Now, I give myself a pat-on-the-back if I simply understand a question !!

Being a physicist I am happy to be correct to the nearest minute! I did it mentally, so don’t have a more accurate figure to hand. So I lose 2 points.

I too enjoy University Challenge. However I am in my local pub at the moment so will have to watch it on catch up.

Pretty good for doing it in your head.

Dozey said

“Being a physicist I am happy … :smile:

… on reflection :sunglasses:, I think even a physicist should be able to figure out the equivalent of 40m at 65mph using mental arithmetic… :upside_down_face:

The vulgar fraction option is easier to derive in your head rather than the decimal version. But of course, typing a vulgar fraction into an i-phone after a few pints in a pub … well, that’s a Brain Teaser in itself !! :laughing:

Mental Arithmetic Solution

The time is clearly between 4 o’clock and 5 o’clock.

At 4 o’clock the hour hand is over the 20 minute mark, whilst the minute hand is over the 60 minute mark.

The distance between the two hands is 60 – 20 = 40 minutes (measured anti-clockwise from the hour hand) – But think of it as “Miles” - if this helps, ie assume it’s a VERY large clock !

The speed at which the minute hand travels is 60mph (minutes per hour, but think of it as miles per hour if this helps !)

The speed at which the hour hand travels is 5mph (using the minute marks on a clock face. But again, think of it as miles per hour if that helps)

The two hands are moving towards each other at a closing speed of 65mph (ignoring relativity effects etc) along a track that is 40m long.

It will take 40/65 hours before they meet.

Or, 60x(40/65) minutes.

Now for the mental arithmetic ……

Divide Top and Bottom by 5

Gives 60x8/13

Or 480/13

Can you divide 480 by 13 in your head ?

You should get 36 with 12 remaining

Answer = 36and12/13 minutes after 4 o’clock

If you used a calculator, you will have something like 36.923……minutes after 4 o’clock.

Which is not quite so elegant !

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The Octagon No 2

As before, there are three empty rectangles.

The symbols +, - , and x (Plus, Minus and Multiply) are to be placed, one in each of the empty rectangles to make a sum that equals the number in the central rectangle, ie 32.

Each symbol must be used, and the calculation is made in the direction of the arrows, ie clockwise.

5 x 6 - 10 + 12 = 32

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I like the relative speed approach. Very elegant.

Thank you Dozey.

Stairway to Heaven.

Above is a drawing showing the finishing stage of a race to the top of a staircase. Three men took part Albert, Bertie and Charlie.

Staring from the bottom, Albert went up three risers at a time. Bertie did four at a time and Charlie took five risers each stride !

Albert obviously wins, as can be seen. But the question isn’t about who wins, it’s about how many risers are there in the staircase ?

I’ve only shown the top of the staircase. I didn’t want to draw the whole staircase – it wouldn’t be a Brain Teaser if I did ! There could be tens or hundreds of risers below those shown, or just a few.

But it is possible, from the evidence presented, to determine the fewest possible risers in the staircase.

Anybody ?

My first thought when I encountered this teaser, was 60. Because that was the lowest multiple of 3, 4 and 5.

Do we assume that each of them reached the top with exact numbers of paces Don?

Hi Eoink,

That is a very good question. The basic answer is “No !”

The clue is in the very first sentence of the Brain Teaser … “Above is a drawing showing the finishing stage of a race to the top of a staircase.”

I have copied the sentence and highlighted a few key words.

By inference, each person started their climb using 3, 4 or 5 risers as allocated, but the diagram shows the actual finishing stage of the climb.

We can see that A will have only one step to finish, B will have 3 steps to finish, etc.

My interim reference to 60 being the lowest multiple of 3, 4, 5 was intended to make it clear this was not the correct answer, and that a somewhat closer inspection of the wording and the diagram was required.

It seems to have worked !