“I’m proud that I’ve lived to see… so many of the things that I’ve worked on being so widely adopted that no one even thinks about where they came from.” Solomon Golomb (1932-2016)
Solomon Golomb, who died on Sunday May 1st, was a man who revelled in the key objects in a recreational mathematician’s toolbox: number sequences, shapes and words (in many languages). He also carved out a distinguished career by, broadly speaking, transferring his detailed knowledge of the mathematics behind integer sequences to engineering problems in the nascent field of digital communications, and his discoveries are very much still in use today.
Puzzlebomb is a monthly puzzle compendium. Issue 53 of Puzzlebomb, for May 2016, can be found here:
Puzzlebomb – Issue 53 – May 2016
The solutions to Issue 53 can be found here:
Puzzlebomb – Issue 53 – Solutions – May 2016
Previous issues of Puzzlebomb, and their solutions, can be found at Puzzlebomb.co.uk.
The first puzzle is a super-fun 25×25 nonogram puzzle
Before Christmas, the benign megasurveillance bods at GCHQ released a set of festive puzzles, in the form of a Christmas card and associated website. An initial nonogram puzzle led to a sequence of increasingly fiendish teasers, and solvers of the final set of puzzles were invited to email in their answers, with the correctest winning a fancy paperweight, signed book and, GCHQ were at pains to stress, not an Imitation-Game-style secret job offer.
Let’s play a game:
- Imagine you have some playing cards. Of course if you actually have some cards you don’t need to imagine!
- Pick your favourite natural number $n$ and put a deck of $n$ cards in front of you. Then repeat the next step until the deck is empty.
- Take $2$ cards from the top of the deck and throw them away, or just take $1$ card from the top and throw it away. The choice is yours.
If you pick a small $n$, such as $n=3$, it’s pretty easy to see how this game is going to play out. Choosing to throw away $2$ cards the first time means you’re then forced to throw away $1$ card the next time, but only throwing away $1$ card the first time leaves you with a choice of what to throw away the next time. So for $n=3$ there are exactly $3$ different ways to play the game: throw $2$ then $1$, throw $1$ then $2$, or throw $1$ then $1$ then $1$.
Now, here comes the big question. How does the number of different ways to play this game depend on the size of the starting deck? Or in other words, what integer sequence $a_0$, $a_1$, $a_2$, $a_3$, $a_4$, … do we get if $a_n$ represents the number of different ways to play the game with a deck of $n$ cards? (We already know that $a_3=3$.)
If you’ve worked with or used any sequences of integers lately (and let’s face it, you have) you might have looked them up in the OEIS. I’ve used it twice today, and it’s still before 9.30am. As you may have gathered from our extensive banging on about it, we’re huge fans of the Online Encyclopedia of Integer Sequences.
If you have visited their site recently, you might have noticed an extra paragraph of red text near the top – yes, they’re doing a Wikipedia, and asking for their users (which is realistically everyone) to donate so they can keep going. It’s a hugely worthy cause, and here at the Aperiodical, we think it’s worth supporting. The OEIS is owned and maintained by The OEIS Foundation Inc., a nonprofit company.
Head over to the OEIS for lists of integers with various properties, and to find out more.
Radio 4 maths police More or Less took time off from calling out journalists and deputy prime ministers for their misuse of statistics this series to sneak a hidden maths puzzle into their show. The first five episodes were “brought to us by” the numbers, respectively, 1, 49, 100, 784 and 1444. Listeners were invited to work out what number would bring us the final episode.