Aesop's Fables... in Latin!
Synchronicity: just last night I was showing somebody this game AND this morning someone wrote and asked me to explain it to her again... which is the perfect excuse to publish it here.
Just like we do, the Romans counted on their fingers. The Roman numerals I, V and X are NOT letters. They may look like letters... but they are not! They are symbols for fingers of the hand and/or for notches made on sticks (compare our tally system of IIII with the fifth mark being a diagonal through the group, and then starting a new group of five next to it). So, you can think about it this way:
I is one finger
II is two fingers
III is three fingers
IIII is four fingers
V can be seen as five fingers (a schematic outline view of "V" made by a hand with all fingers extended; see below  it just looks like the letter V)
X is two sets of five, one above and one below (it just looks like the letter X, but it's really two Vs, one up and one down)
The use of "IV" to indicate 4 came later, and IIII continued to be used as a representation of the number 4 (it makes sense, doesn't it?).
Now learn how to multiple numbers 5 and higher using finger counting! Here is an illustrated guide to the Roman counting game which lets you multiply numbers 59 on your fingers. Showing the numbers 1, 2, 3, or 4 with your fingers is easy; you just hold up that number of fingers. To be able to do this trick, you also need to know how to show the numbers 6, 7, 8 and 9 with ONE HAND ONLY, being able to show those numbers with either your left or your right hand. Here is how it goes:
LEFT  RIGHT


To show the number 5, hold no fingers down, palm facing you, so all fingers are extended; here is how the number 5 looks when made with your left hand, and with your right hand: 

To show the number 6, hold one finger down, palm facing you; here is how the number 6 looks when made with your left hand, and with your right hand: 

To show the number 7, hold two fingers down, palm facing you; here is how the number 7 looks when made with your left hand, and with your right hand: 

To show the number 8, hold three fingers down, palm facing you; here is how the number 8 looks when made with your left hand, and with your right hand: 

To show the number 9, hold four fingers down, palm facing you; here is how the number 9 looks when made with your left hand, and with your right hand: 
To multiply, do the numbers with both hands, and then COUNT the number of fingers down and multiply those by 10, and then MULTIPLY the number of fingers standing up, and then add the two numbers:
6 multiplied by 7: 3 fingers down = 30 

9 multiplied by 9: 8 fingers down = 80 

7 multiplied by 8: 5 fingers down = 50 

8 multiplied by 5: 3 fingers down = 30 
Try some more  it works!
I learned this trick in a medieval Latin graduate seminar, and I recall somehow that the source was the venerable Bede. Does anybody know exactly where to find that in Bede? I'd love to know more about how math was done with Roman numbers in general  if anybody has reading suggestions for that, online or in print, please leave a comment here! :)
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OK. There's how it works: First of all some notation. Fingers down on one hand will be D1 and fingers down on the other hand will be D2. To show a number five and up on either hand you lower a finger. Increasing the amount of lowered fingers increases the value of the hand. The product of two numbers using the notation above looks like this: (5 + D1) (5 + D2), which, multiplied out, gives
25 + 5D1 + 5D2 + D1D2 (Equation 1)
Now the method of reading the result combines notions of fingers down and fingers up, which are related like this (U = "up"): U1 = 5  D1 and U2 = 5  U2.
The product is supposedly the sum of downed fingers times 10 plus the product of the up fingers, using our notation:
10 * (D1 + D2) + (5  D1) (5  D2),
which, multiplied out, gives
10D1 + 10D2 + 25  5D1  5D2 + D1D2
which is equivalent to
5D1 + 5D2 + 25 + D1D2 (Equation 2)
Comparing these two expressions (Equation 1) and (Equation 2) shows that they are identical quantities.
QVOD ERAT DEMONSTRANDVM (whoo hooo...get to USE that!)
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