For example, in the sequence 1, 2, 4, 8, 16, … each term is exactly 2/3rds of the previous term, plus 1/3rd of the following term. What other sequences have exactly that property?

For a given value p, what sequences s₁, s₂, s₃, … s_n have the property that each s_k = p s_(k-1) + (1-p) s_(k+1) ? Does knowing s₁ also fix the remaining terms? 

Amusingly, this actually will help with last weeks puzzle!

Standing at 4:00, repeatedly flip an unfair coin, moving clockwise if it comes up heads, and counterclockwise if it comes up tails.

What is the probability that you visit every hour 1-11, before you reach 12:00?

In our next post we’ll ask a seemingly unrelated question that turns out to help quite a bit…

This made me smile.  I hope it makes you smile too.

What is

If you think they are the same then why?  If you think they are different then why?

First, a warm-up puzzle:

You’re blindfolded and I will place two cards on the table in front of you, each either face up or face down at my option.  We’ll then play a game taking turns.  On your turn, you may turn over either one or both cards.  On my turn, I may either swap the positions of the cards without changing which might be face up or not at my option.  You are unable to detect whether or not I’ve made a swap.  You win as soon as both cards are face up.  Your task is to identify a set (deterministic) strategy that you can use such that regardless of how I place and switch cards you are sure to win.

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Now, consider a tougher version of the problem.  You now have four cards, each on the corner of a square table.  The setup is the same as before.  You’re blindfolded and I place the cards faces up or down in a way that I think will stump you.  You flip 1,2,3 or all cards on your turn, and then I either leave the table alone or rotate it 90, 180 or 270 degrees.  You can’t tell if the table has turned.  You then do some more flipping, etc.

(Note that I can’t arbitrarily re-order cards — only rotate the table.  Their relative position remains constant.  I didn’t labor this point in the first puzzle as with two cards it isn’t a distinction.  You can imagine that the rules are identical and the table can only be rotated 180 degrees.)

Can you identify a pattern of steps you can take that will ensure victory regardless of my placements and turning?

I am in a total immersion game called ‘World of Britain’.

Every day you can take a daily task, if you dare.  There are five tasks and each day one of these tasks is given at random.  You could get the same task each day if you were unlucky.

The tasks are:

·         Cycling in Edinburgh; the best city in the world has plenty of cycle lanes to help you avoid the traffic, some of them are a bit surprising.

·         Cheese Rolling in Gloucestershire; can you beat the cheese? 

·         Bog Snorkelling in Wales; my favourite British sport, bar none!  Watch the action here!  

·         Gurning goes back to at least the thirteenth century.  You may think it is just about pulling funny faces, and you would be right.

·         Mud Racing  How do you know who won?

My math question is:  assuming you win everything you enter how many days would you expect it to take to win all of these competitions?

p.s Gurning is now international.  Here is a US gurner completely oblivious to the exciting belly-flop championship happening behind him.  Apparently “everyone and their butt crack is welcome”

Thanks to my second favourite math radio programme, More or Less, for this wonderful new insight into music and maths.

In her song, Pi, Kate Bush sings the first one-hundred and fifty or so digits of the celebrity number. 

This is what she sang:

You don’t need me to point out the wrong digits do you?  Good.  Then we can move on.

This has led to the Kate Bush Conjecture.  Since Pi contains an infinite sequence of digits which never repeat surely the sequence Kate sings must occur somewhere!  She never says she is starting at the beginning. 

The Weak Kate Bush Conjecture says:

                The sequence Kate sings exists somewhere in the decimal expansion of Pi.

The Strong Kate Bush Conjecture says:

                Kate could have sung any finite sequence of digits and it would exist somewhere in the decimal expansion of Pi.

If Pi were a random sequence of digits then both conjectures are true.  But Pi isn’t random, it is a well-defined number so we can’t make any assumptions.  Instinctively I think it must be true, but that isn’t good mathematics, we need to prove it!

For example the following number is infinite and non-repeating but it doesn’t satisfy either conjecture:  0.01001100011100001111…

If the strong conjecture is true then every finite sequence exists in Pi.  And they each exist an inifinite number of times since they can occur in an infinite number of longer sequences.  Think about that, an infinite number of sequences each occurs an infinite number of times.

Everything that can be encoded digitally would occur within Pi.  That would include the complete works of Shakespeare, naturally, and also the note you left for the milkman last Tuesday, and those poems you wrote when you were five.

Every religious book, all cannons and all translations, both forwards and backwards.  Every prayer, every satanic chant and every children’s song.

That picture of the cosmic microwave background, the observations of Tycho Brahe and all of Kepler’s notes; and the results from CERN that will prove the Higgs-Boson (come on guys!)

It would include everything on your iPod, every episode of Math Factor and an alternative Math Factor with Groucho Marx.

Every album by Bob Dylan, or Kate Bush.  And everything you’ve sung in the shower.

It would include this post and all the comments you will make, or think about making.

It would include every thought and idea you have ever had, or ever will have, or ever could have.  (Gödel may have something to say about that)

And you thought Kate Bush was just a singer.

And you thought Pi was just a number.

How many times did Arp and Bif shake hands?

(Well, that’s not quite enough information, all you need to know is the exact number of guests — and it turns out that there are only two possibilities in any case.)

We also answer last weeks puzzle and ‘fess up about the cards in the dark problem.

The sum of the digits of 351 is 9 and the sum of the digits of 2 x 351 = 702 is also 9.

1) If a number has this property, can we always rearrange its digits and obtain another number with this property (513, 135, etc all have it)

2) Which powers of 2 have this property?

3) And most of all, can you give a simple characterization of the numbers with this property, in terms of just the digits themselves?

You may draw one ball (at random) out of a box of your chosing.  After that, you must rearrange the labels so that they correctly reflect the contents of the boxes.  How do you do this?