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Science puzzles

Win the lottery -- Friction -- Supersmooth hump -- Reverse statistics -- Ultrashort colour -- First things... -- Entropic ambiguity -- Interstellar speeding

Some of these problems are straightforward calculating exercises, others have to be approached with a mind so open it practically has negative curvature. Good science requires approaching a problem in an unprejudiced way, without making unwarranted assumptions and letting habit get in the way of objectivity. In this spirit, can you solve the following puzzles?

(Well, I admit some of them are plain nasty...)

Win the lottery

You play the lottery. Every week you pick six numbers from 1 to 49 inclusive. Subsequently, six such numbers are drawn by the lottery, and your winnings are higher the more numbers you get right.

Let us assume that the lottery is perfectly random and that the drawing is done in a perfectly objective way which you cannot influence.

How can you maximise your expected winnings?

Solution

The power of friction

You want to lower a heavy object over the edge of a cliff on a rope. In order to lighten the load, you wrap the rope several times round a smooth cylindrical object which is fastened so that it can neither move nor rotate (Let's say, the handrail at a viewpoint). The rope will tighten around the cylinder, and the ensuing friction will reduce the force you have to hold.

How large is the force with which you have to pull at your end of the rope, depending on the load and the number of turns of the rope round the cylinder? The number of turns should be treated as a real number, since you can wrap the rope 3/4 of the way round.

Solution

Supersmooth hump

You want to modify a real function, any real function, at and around a few points, but leave it completely unchanged outside the neighbourhoods of those points. You do not want to affect the function's differentiability in any way - it may be, and should remain, differentiable continuously infinitely many times.

You can achieve that by adding a "hump" function which you scale and shift to fit in the neighbourhoods and to give the value you want. It has to be constant =0 outside an interval and differentiable continuously infinitely many times. Find such a function.

Solution

Reverse statistics

Can you determine the next two elements of the sequence which starts like this?

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

And what does the title of this problem have to do with it?

Solution

Ultrashort colour

Recently physicists have managed to build "attosecond lasers", lasers which emit pulses 10-18 seconds long. Before them, lasers emitting femtosecond (10-15 seconds) pulses have been around. What colour is the light they produce?

Solution

First things first

Write down the first 30 or more powers of two in decimal. Which digit occurs most frequently as their first digit? What is the answer if you write them in octal?

Explain this, and predict the result for the powers of any number written in an arbitrary base.

Solution

Entropic ambiguity

C. E. Shannon defined the information theoretic entropy as:

- Σi p(xi) log2(p(xi))

where xi is a data value which may be transmitted over a channel and p(xi) is the probability with which xi occurs. The unit of the entropy is "bits".

Now consider the following stream of bits, which repeats over and over again:

... 00 01 10 11   00 01 10 11   00 01 10 11 ...

If we define a data word to be equal to a bit, it can assume two possible values which occur with equal frequency, so the entropy would be

- 1/2 log2(1/2) - 1/2 log2(1/2) = 1 bit

If we take each data word to have two bits, four different values occur, with probability 1/4 each, so we get an entropy of 2 bits!

If we take 4-bit data words, we have again two possible values with equal probability, so an entropy of 1 bit. If we define a data word as being 8 bits, only one value occurs, so we obtain an entropy of zero bits.

Which is the real value of the entropy of this data stream?

Solution

Interstellar speeding

Astromoners have observed celestial objects at a distance of 450 million light years which moved by 0.004 arc seconds in 16 months. How can this be?

Solution