I am interested in puzzles and games and how they relate to mathematical thinking, not least through my involvement with the Maths Arcade initiative. I was pleased to read what is said on this topic in the 1982 Cockcroft report. This is the report of an inquiry started in 1978 “to consider the teaching of mathematics in primary and secondary schools in England and Wales, with particular regard to its effectiveness and intelligibility and to the match between the mathematical curriculum and the skills required in further education, employment and adult life generally”.
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- 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.
Puzzlebomb – August 2014
Puzzlebomb is a monthly puzzle compendium. Issue 32 of Puzzlebomb, for August 2014, can be found here:
Puzzlebomb – Issue 32 – August 2014
The solutions to Issue 32 can be found here:
Puzzlebomb – Issue 32 – August 2014 – Solutions
Previous issues of Puzzlebomb, and their solutions, can be found here.
Carnival of Mathematics 112
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of June, and compiled by Robin Whitty, is now online at Theorem of the Day.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Puzzlebomb – July 2014
Puzzlebomb is a monthly puzzle compendium. Issue 31 of Puzzlebomb, for July 2014, can be found here:
Puzzlebomb – Issue 31 – July 2014
The solutions to Issue 31 can be found here:
Puzzlebomb – Issue 31 – July 2014 – Solutions
Previous issues of Puzzlebomb, and their solutions, can be found here.
Carnival of Mathematics 111
The next issue of the Carnival of Mathematics, rounding up blog posts from the month of May, and compiled by Peter Krauzberger, is now online at Boole’s Rings.
The Carnival rounds up maths blog posts from all over the internet, including some from our own Aperiodical. See our Carnival of Mathematics page for more information.
Puzzlebomb – June 2014
Puzzlebomb is a monthly puzzle compendium. Issue 30 of Puzzlebomb, for June 2014, can be found here:
Puzzlebomb – Issue 30 – June 2014
The solutions to Issue 30 can be found here:
Puzzlebomb – Issue 30 – June 2014 – Solutions
Previous issues of Puzzlebomb, and their solutions, can be found here.
Discovering integer sequences by dealing cards
Let’s play a game:
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 $a_3=3$.)