# Inspirational Games

by on October 26, 2012

Arithmetical games are a great way to sharpen up your arithmetic. My Grandad taught me two classic games that gave me a familiarity with numbers before I was even old enough to go to school. These two games are Fives and Threes which is a dominoes game and Cribbage which is a card game.

Fives and Threes
Fives and Threes is played with a peg board to keep track of the score of each player. It should be played with a set of ‘Double Nine’ dominoes, as this offers a much more interesting range of number possibilities. As usual with domino games, players take turns to build a chain of dominoes. But the distinctive feature of Fives and Threes is that players score points depending on the values of the exposed ends of the domino chain. The amount scored is equal to the number of times that the sum of the two ends can be exactly divided by three or five. For instance, if one end of the chain shows a 2 and the other end shows a 4, then the total is 6, which is exactly divisible by 3 twice, so the person who played the previous domino scores 2. The highest scoring totals are those that are simultaneously divisible by 5 and 3. For instance, if one end of the chain is 7 and the other end is 8, this produces a total of 15, which is divisible by both 5 and 3. It is divisible by 5 three times and it is divisible by 3 five times, so this gives a score of 3 + 5 = 8.

The full rules of Fives and Threes are given on the following website:
http://www.pagat.com/tile/wdom/fives_and_threes.html

Cribbage
The card game Cribbage is also played with a peg board to keep track of the scores. Points are scored for card combinations that add up to fifteen, and for pairs, triples, quadruples, runs and flushes. As well as the obvious arithmetical task of finding cards whose values sum to 15, Cribbage is of interest because of the importance that combinations and permutations have in the game. For instance, a pair of numbers with the same value, say two 7s, would be worth two points. A triple, say three 7s, would be worth six points, because there are three ways to choose a pair of 7s from the triple, and three times two (the value of a pair) is six. A quadruple of four 7s is worth twelve points, because this gives six ways to choose a pair of 7s, and six times two is twelve.

Points are also awarded for every combination of cards that sum to 15. For instance, a hand formed of a 6, two 7s and two 8s, can be combined to form 15 in four ways, because each of the 7s can be added to each of the 8s to form 15. Similarly points are scored for every combination of cards that produces a run of three or more cards. For instance, a 6, two 7s and two 8s, would produce four runs of three cards.

This may sound a bit complicated if you are unfamiliar with Cribbage, but it is quite straightforward really and definitely worth investigating. The full rules of Cribbage are given on the following website:

According to John Aubrey, Cribbage was invented by the aristocratic English poet Sir John Suckling early in the 17th century.

http://en.wikipedia.org/wiki/John_Suckling_(poet)

Both Fives and Threes and Cribbage are traditional British pub games, but they are great games nonetheless. Cribbage remains my favourite card game.

The Richmomachia board at the start of a game.

Rithmomachia
The most elaborate of all arithmetic games must be Rithmomachia or The Philosophers’ Game. The name Rithmomachia means ‘Battle of the Numbers’. Descriptions of the game survive in manuscripts from as long ago as the 11th and 12th centuries, when it seems to have been played in Benedictine monasteries. Rithmomachia is like an arithmetical relative of Chess. Each player has a number of circular, triangular and square pieces, as well as a single pyramid. These pieces have numerical values. Like Chess, play involves the capture of the pieces belonging to the opponent, but unlike Chess the success of an attack depends on the number combinations of the pieces.

The game is reputed to have been invented as a method of teaching Pythagorean numerology, as set out in The Consolation of Philosophy written by the fifth century poet Boethius – a book that was very highly esteemed throughout the European Middle Ages. Rithmomachia was highly regarded by leading late medieval British philosophers and mystics, such as Roger Bacon, Thomas More and John Dee.