Chapter 6 — Number Play

Number Play explores five areas: reading information encoded in number sequences, parity (odd and even rules for sums), number grid puzzles and magic squares, the Virahāṅka–Fibonacci sequence and its origins in Indian poetry, and cryptarithms where digits are replaced by letters.

Parity refers to the property of a number being even or odd. An even number can be arranged in pairs with no leftovers; an odd number is one more than a collection of pairs. The chapter uses this idea to determine whether sums, differences, and products are even or odd without calculating the exact value.

Even + even = even. Odd + odd = even. Even + odd = odd. The sum of any number of even numbers is always even. Adding an odd count of odd numbers gives an odd result; adding an even count of odd numbers gives an even result.

Adding five odd numbers (an odd count of odd numbers) always gives an odd result. Since 30 is even, it is impossible for five odd numbers to sum to 30, regardless of which odd numbers are chosen.

A magic square is a grid where each row, each column, and each diagonal all add up to the same number, called the magic sum. For a 3×3 grid using the numbers 1–9, the magic sum must be 15, because the numbers 1 through 9 sum to 45, and that total is shared equally among 3 rows.

The centre must always be 5. Numbers like 1 and 9 cannot be at the centre because there is no way to pair them with two other available numbers from 1–9 to reach the required row sum of 15. By elimination, only 5 works at the centre.

The Virahāṅka–Fibonacci sequence is 1, 2, 3, 5, 8, 13, 21, 34, … where each term equals the sum of the two before it. It was first explicitly described by the Prakrit scholar Virahāṅka around 700 CE while counting the number of rhythms with a given number of beats, where a short syllable lasts 1 beat and a long syllable lasts 2 beats.

The Italian mathematician Fibonacci wrote about these numbers in 1202 CE, but Virahāṅka had already described them around 700 CE — about 500 years earlier. Earlier Indian contributors include Piṅgala (c. 300 BCE), Gopala (c. 1135 CE), and Hemachandra (c. 1150 CE). The term 'Virahāṅka–Fibonacci numbers' is used so that everyone understands what is being referred to while acknowledging the Indian origin.

The nth even number is given by the formula 2n. The nth odd number is given by 2n – 1. For example, the 100th even number is 200, and the 100th odd number is 199.

The Chautīsā Yantra is the first recorded 4×4 magic square. It is found in a 10th-century inscription at the Pārśhvanath Jain temple in Khajuraho, India. Every row, column, and diagonal in this magic square adds up to 34 — 'Chautīsā' means 34 in Hindi.

Cryptarithms (also called alphametics) are arithmetic puzzles where each digit (0–9) is replaced by a letter. Each letter stands for exactly one digit. You solve them by using place-value reasoning and trial — for example, figuring out which digit T must be when T + T + T gives a two-digit sum whose units digit is also T, leading to T = 5.

The chapter shows that a 3×3 grid using numbers 1–9 must have row and column sums between 6 (= 1+2+3) and 24 (= 9+8+7). If any required circle sum falls outside this range, the puzzle is impossible. For instance, a grid requiring a row sum of 5 or a column sum of 26 has no solution.

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