## Saturday 24 February 2018

### Maths through cards

I wrote this about 20 years ago for the Home Education Network HEN Newsletter. The small girls mentioned grew up straight and tall as Dau.I and Dau.II, so having many packs of playing cards in the home did them no harm. Playing cards was almost the entirety of their math-education! I was shocked to find out that about a third of my current students had never handled a deck of cards: no wonder innumeracy is so prevalent.

I have idled away hours (probably days) of my life playing solitaire or patience.  It's fun to play these, officially one-person games, with small children, although it requires a certain amount of 'patience' to play at a different (at the moment slower) pace.  I look forward to the days, now not far off, when they will be running circles around my dotage. Klondike is the variety that was originally fitted-as-standard in Windows.

There is a deal of maths in a pack of cards.  All those things beyond +, -, *, /, that appear in the children's maths books: Sets, sums, symmetry, equality, counting, logic....  If you think about it, there isn't much left outside of what they call mathematics nowadays.
Most of the games most people play seem to be competitive, even if, like strip-jack-naked or beggar-my-neighbour they involve neither a jot nor a tittle of skill.  I am sure there are well adjusted families who handle winning and losing with equanimity but we find it all rather stressful.  So here's a tuthree games of patience to play with your kids . . . or self if stuck in departures with a pack of cards.

1. Tens.  a. Shuffle a complete pack.  b. Deal out 12 cards.  c. If any of the 12 are 'face' cards (king, queen, jack) remove them and place at bottom of pack.  d. Fill in the holes.  e. Repeat steps c-d until 12  Ace-10 type cards remain.  Now you're ready to start.  Dealing from the top of the pack, cover up any PAIR of exposed cards that add up to 10 or any single 10s: Ace counts as one.  If you get it out, you'll have the 12 face cards exposed and all others covered.  Occasionally, you cannot complete the process, but usually you can get this one out.  No skill or significant decision making required.  After a few games you'll never forget the pairs of numbers that sum to ten.

2. Up and down.  a. Shuffle a complete pack.  b.  Deal out 36 cards face up in a pyramid - one card in the top row, covered by two in the next row, covered by three ... until the last row has seven cards.  c.  From the stock that remains in your hand, turn over the top card on the table in front of you.  d. If the value of one of the wholly uncovered cards is either one higher or one lower than the card in front of you, then take this card from the pyramid and add it to the pile in front of you.  e.  Repeat step d until you can make no more moves, then repeat step c.  It is rare to get this one out.  Some decision making is required.

There was a time when I knew twenty different games of solitaire, some totally mindless, robotic time-passers, some requiring a good deal of foresight and following a logical path many steps long.  Stop me and I'll tell you a few.  It is useful to have a few in your toolkit because playing one game will effectively shuffle the deck w.r.t. another.

For those who can handle a more competitive edge, our once-upon-a-neighbour Roger recommends Cribbage.  This is good for exposing the logic of permutations, matching, and summing to 15 or 31.  Score is usually kept on a cutey marker board with rows of little holes drilled in it [see R].  You can get the rules out of any book of card games.  That same book should have the rules for Piquet, perhaps the most elegant and sophisticated two-handed card game ever played, which is a cross between rummy, whist and medieval french.
A classical deck of cards is just beautiful to look at.  The detailing of the face cards of the traditional "English" pack is beautiful, conventional, stylised and nearly 500 years old.  The other cards give scope for an investigation of  rotational or reflectional symmetry.  Putting aside the abstraction of the pure maths that is in a deck of cards, you can also cover the applied maths of structural engineering when you build card houses.  Every family should have several packs.