Medieval British kings would send accountants to get accurate records of shepherds' holdings of sheep. The shepherds would be instructed by the accountants to repeatedly bring a pair of sheep from pasture into a corral containing two pens and send one of the pair into the first pen and the other into the second pen until there were no more pairs of sheep in pasture. The number of sheep remaining in pasture was recorded by the accountant. What records are possible? Hint || Solution || Private Session || Math Circle
After the initial division of sheep, the accountant would instruct the shepherd to send the unpenned sheep and the sheep in the first pen back to pasture, and then release the sheep in the second pen into the corral to be divided as they were before. After the second division concluded, the accountant would record the number of sheep remaining in the corral to the left of the first number recorded. What records are possible? Hint || Solution || Private Session || Math Circle
This process was repeated with each successive number of sheep in the second pen until it was no longer possible (or necessary). Suppose this went on for n divisions of the sheep. How many records are possible? Hint || Solution || Private Session || Math Circle
What is the number of sheep in terms of n? Hint || Solution || Private Session || Math Circle
Does this system admit place value? Hint || Solution || Private Session || Math Circle
Does this system admit addition, subtraction, multiplication, division, and fractions? Hint || Solution || Private Session || Math Circle
Determine the properties it has in common with decimal. Hint || Solution || Private Session || Math Circle
Discuss the numeration system which would result if the shepherd instead had n pens and separated n sheep at a time (OESCn). Hint || Solution || Private Session || Math Circle
Construct an algorithm for translating from OESCm to OESCn. Hint || Solution || Private Session || Math Circle
For which values of n is OESCn in use today? Hint || Solution || Private Session || Math Circle
In radix b, how does multiplying by bn affect the position of the separatrix? Hint || Solution || Private Session || Math Circle
Construct algorithms for adding, subtracting, multiplying, and dividing numbers in radix b.Hint || Solution || Private Session || Math Circle