Free Professor Stewart's Hoard of Mathematical Treasures by Ian Stewart Page B

big;
    • they preferred even numbers, even when this led to bigger numbers or more of them.
    For example, the computer found that
but both numbers are odd and 703 is large. The scribes preferred
with two even numbers and nothing very big. Gillings gives an extensive discussion in his Mathematics in the Time of the Pharaohs. This book is a bit long in the tooth, and the historical study of Egyptian mathematics has moved on, but it still has a lot of interesting things to say.

The Greedy Algorithm
    Egyptian fractions are obsolete for practical arithmetic, but still very much alive as mathematics, and you can learn a lot about modern fractions by thinking about Egyptian ones. For a start, it’s not obvious that every fraction less than one has an ‘Egyptian representation’ - as a sum of distinct unit fractions - but it’s true. Leonardo of Pisa, the famous ‘Fibonacci’ (Cabinet, page 98), proved this in 1202, showing that what is nowadays called the ‘greedy algorithm’ always does the job. An algorithm is a specific method of calculation that always produces an answer, like a computer program.
    The greedy algorithm begins by finding the largest unit fraction that is less than or equal to the fraction you want to represent - that’s what makes it greedy. Subtract this fraction
from the original fraction. Now repeat, looking for the largest unit fraction that is different from the one you got the first time, but less than what’s left. Keep going.
    Amazingly, this method eventually reaches a unit fraction and stops.
    Let’s try out the greedy algorithm on the fraction
    • Find the biggest unit fraction that is less than or equal toThis is
    • Find the difference
    • Find the biggest unit fraction different fromthat is less than or equal toThis is
    • Find the difference:
    • Find the biggest unit fraction different fromandthat is less than or equal toThis isitself, so the algorithm stops.
    Putting the pieces together, we have
which is the required Egyptian representation.
    The greedy algorithm doesn’t always give the simplest Egyptian representation. For instance, when applied toit produces
and fails to spot a simpler answer:
    The Erdős-Straus conjecture states that every fraction of the form 4/n has a unit fraction representation with three terms:
It is true for all n < 10 14 . Exceptions, if they exist, must be very thin on the ground, but no proof or disproof exists.
    There are also some interesting variations on the greedy algorithm that you can try. I suggest using fractions with small numerators and denominators to avoid monsters like the one we’ve just seen. First, try it with the extra condition that every fraction involved must be one over an even number. Surprisingly, the greedy algorithm still works - it has been proved that every fraction less than one is a sum of unit fractions with distinct even denominators.
    Now try it with odd denominators. Computer experiments suggest that it also works in this case. For instance,
But now nobody has a proof. For all we know there might be some peculiar fraction for which the odd-denominator greedy algorithm goes on for ever.
    Now that’s really greedy.
    We have only scratched the surface of the mathematics of Egyptian fractions. For more, see: en.wikipedia.org/wiki/Egyptian_fraction

How to Move a Table

    William Feller.
    William Feller was a probability theorist at Princeton University. One day he and his wife wanted to move a large table from one room of their house to another, but, try as they might, they couldn’t get it through the door. They pushed and pulled and tipped the table on its side and generally tried everything they could, but it just wouldn’t go.
    Eventually, Feller went back to his desk and worked out a mathematical proof that the table would never be able to pass through the door.
    While he was doing this, his wife got the table through the door.

Rectangling the Square
    Form five rectangles by choosing their sides from the list 1,