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

the cards into the same order, deals them into 4 rows of 13 cards, and again asks which row the card is in.
    After which he unerringly names the chosen card.
    How does this trick work?
     
    Answer on page 293

Halloween = Christmas
    Why do mathematicians always confuse Halloween and Christmas?
     
    Answer on page 293

Egyptian Fractions
    Whole numbers are fine for addition and multiplication, but subtraction causes problems because, for instance, 6 - 7 doesn’t work with positive whole numbers. This is why negative numbers were invented. A positive or negative whole number is called an integer.
    In the same way, the problem of dividing one number by another, such as 6 ÷ 7, 18 requires the invention of fractions like. The number on the top (here 6) is the numerator, the one on the bottom (here 7) is the denominator.
    Historically, different cultures handled fractions in different ways. The ancient Egyptians had a very unusual approach to fractions; in fact, they had three unusual approaches.
    First, they had special hieroglyphs forand

    Hieroglyphs forand
    Second, they used various portions of the Eye of Horus, or Wadjet Eye, to represent 1 divided by the first six powers of 2.

    Wadjet Eye (left), and fraction hieroglyphs derived from it (right).
    Finally, they devised symbols for fractions of the form ‘one over something’, that is,,,,, and so on. Today we call these
unit fractions. The unit fraction 1/n was represented by placing a cushion-shaped hieroglyph (normally representing the letter R) over the top of the symbols for n.

    Hieroglyphs for 1/1,237
    (in practice the Egyptians wouldn’t have used such big numbers in a unit fraction).
    However, these methods dealt only with special types of fractions, and 6 divided by 7 was still a problem. So the Egyptians expressed all other fractions as sums of distinct unit fractions, for instance
and
It’s not at all clear why they didn’t like to writeas+, but they didn’t.
    Doing arithmetic with unit fractions is weird, but possible. Our method is very different: we ‘put both fractions over a common denominator’ (page 310) like this:
We can see that the result is roughly 1, which isn’t obvious from the Egyptian fractions.
    Nevertheless, the Egyptians did amazing things with their symbolism. Our most important source for their work is the Rhind mathematical papyrus, now in the British Museum. Alexander Rhind bought the papyrus in 1858 in Luxor; it seems
to have been unearthed by unauthorised excavations near the Ramesseum.

    Part of the Rhind mathematical papyrus.
    The papyrus dates to around 1650 BC, in the Second Intermediate Period. The scribe Ahmose copied it from an earlier text from the time of the 12th dynasty pharaoh Amenemhat III, two centuries earlier, but the original text is lost. It measures 33 cm by 5 m, and even now scholars do not understand everything on it. However, one remarkable section, about one-third of one side, deals with unit-fraction representations of numbers of the form 2/n, where n is odd and runs from 3 to 101.
    Ahmose’s results here can be summed up in a table. To simplify the printing and improve legibility, an entry like
means that
The table is impressive, but also raises a number of questions. How did whoever first found these representations discover
them? Why did the scribes prefer these particular representations?
    Expressing 2/n, for n odd, as a sum of at most four unit fractions.

    In 1967, at the request of Richard Gillings, C. L. Hamblin programmed an early electronic computer belonging to Sydney University to list all possible ways to represent the fractions 2/n in Ahmose’s table as sums of unit fractions. The results led Gillings to argue that:
    • the Egyptians preferred small numbers;
    • they preferred sums with two unit fractions to those with three, and sums with three unit fractions to those with four;
    • usually they liked the first number to be as small as possible, but not when that made the last number too