Other ways to calculate, and harder puzzles

You can already add, subtract, multiply and divide numbers as big as crores, and you know how to check your answers. So is there anything left to explore? A great deal.

The column methods you learned at school are only one way to calculate. For centuries people in India, the Arab world, China, Egypt, Russia and Europe used other methods: filling in a grid of diagonals, laying out carved rods, doubling and halving, sliding beads on an abacus. Each method works for the same deep reason (place value and the way multiplication spreads over addition), but each one looks completely different.

This layer is a playground. You will try other ways to calculate, crack puzzles that have entertained people for hundreds of years, tackle olympiad-style problems, plan real projects with real budgets, and meet people whose jobs depend on the four operations. There is no single right order. Dip in wherever looks interesting.

Here is a method that feels like drawing, not calculating. It is called lattice multiplication, or the gelosia method (a gelosia is an Italian window screen with a diagonal criss-cross pattern, which is what the grid looks like).

To multiply 473 × 58, draw a grid with one column for each digit of 473 and one row for each digit of 58. Write 473 along the top and 58 down the right-hand side. Draw a diagonal line through every box, from its top-right corner to its bottom-left corner.

In each box, write the product of its column digit and row digit, with the tens above the diagonal and the ones below it. So 4 × 5 = 20 is written as 2 / 0. Every box holds a two-digit answer (a single digit gets a 0 in front: 2 × 3 = 06).

Napier's bones are a set of rods, one for each digit 0 to 9. Each rod shows that digit's times table, written in little diagonal-split squares exactly like a lattice. To multiply 473 × 8, you lay the 4, 7 and 3 rods side by side, look along the row for 8, and add along the diagonals. Scottish mathematician John Napier published the idea in 1617. The rods were popular across Europe, and sets were even made in China and Japan. They are really a lattice you can carry in your pocket.

In India you may have seen books and classes on Vedic Maths, full of fast tricks with Sanskrit names. The tricks are real, clever and fun, and every one of them works because of place value. Here are four of the best, each with the reason it works. Treat them as extra tools for special numbers, not as a replacement for the general methods.

Squaring numbers that end in 5. Take the tens part, multiply it by the next number up, and write 25 after it. 35 × 35: 3 × 4 = 12, then 25, giving 1225. 85 × 85: 8 × 9 = 72, giving 7225. It even works for 105 × 105: 10 × 11 = 110, giving 11,025.

Why? 35 × 35 = 30 × 35 + 5 × 35. Split the second part: 5 × 35 = 5 × 30 + 5 × 5. Now 30 × 35 + 5 × 30 = 30 × (35 + 5) = 30 × 40 = 1,200, and 5 × 5 = 25. So 35 × 35 = 30 × 40 + 25. The 'tens times next number' is just 30 × 40 in disguise.

Imagine you only knew how to double, halve and add, with no times tables at all. Could you still multiply? Yes. The ancient Egyptians multiplied this way. The Rhind papyrus, which the scribe Ahmes copied about 1550 BCE from an older document of the Middle Kingdom, shows the method; the same idea is known today as Russian peasant multiplication.

To find 37 × 24: write the two numbers at the top of two columns. Keep halving the left number (throw away any half) and doubling the right number, until the left column reaches 1. Then cross out every row where the left number is even, and add what remains in the right column.

An abacus is a frame of rods with sliding beads, one rod per place value. On the Japanese soroban, each rod has one 'heaven' bead worth 5 and four 'earth' beads worth 1 each, so a rod can show any digit 0 to 9. Adding means pushing beads towards the bar; when a rod would pass 9, you clear it and push one bead on the next rod to the left. That is carrying, done with your fingers.

Children who train on an abacus for a long time often learn to picture it in their heads and slide imaginary beads, adding long lists of numbers astonishingly fast. Abacus and mental-arithmetic classes are popular in many Indian cities today.

India also produced one of the most famous mental calculators of all time: Shakuntala Devi (1929–2013), often called the 'human computer'. On 18 June 1980, at Imperial College London, she multiplied two 13-digit numbers in her head and gave the correct 26-digit answer in 28 seconds — a feat recorded in the 1982 Guinness Book of World Records.

Here is a puzzle that has been told, in different versions, for about a thousand years in India, the Arab world and Europe:

A woman takes her eggs to market. When she puts them in rows of 2, one egg is left over. The same happens with rows of 3, 4, 5 and 6: always one left. But rows of 7 fit exactly. What is the smallest number of eggs she could have?

Try it before reading on. Hint: first find numbers that leave remainder 1 for all of 2, 3, 4, 5 and 6.

Sums of consecutive numbers. 15 can be written as a sum of two or more consecutive whole numbers in three ways: 7 + 8, 4 + 5 + 6 and 1 + 2 + 3 + 4 + 5. Try some other numbers: 9 = 4 + 5 = 2 + 3 + 4. 10 = 1 + 2 + 3 + 4.

Now try 8. Or 16. Or 32. You will not manage it. A computer search of 1 to 64 shows the only numbers that cannot be written this way are 1, 2, 4, 8, 16, 32, 64: exactly the powers of 2. Proving why is a lovely olympiad challenge. (A clue: a run with an odd number of terms is that many copies of its middle number, so its total has an odd factor.)

The best way to master the four operations is to use them for something that matters. Each project below is a real task: you gather numbers, choose operations, estimate, calculate and present a decision. Work alone, in a pair or as a class.

For the kirana project, visit a local shop with an adult (or use a real bill from home). Record quantities and rates, then calculate each amount and the total before you look at the printed bill. In the sample above, the total is ₹1,383, and paying with ₹2,000 gives ₹617 change. Then extend it: what if every price rose by ₹5? What if you bought the same things every week for a year (52 weeks)? The yearly bill would be 52 × ₹1,383 = ₹71,916.

India's 2011 Census counted 1,21,08,54,977 people (1,210,854,977 in the international system). Numbers like this come with their own questions, and every one needs the four operations:

• How many per square kilometre? Divide population by area. This is population density.
• How much did it grow? Subtract one census count from the next, then compare with the earlier count.
• How many per polling booth, per school, per doctor? Divide, then decide whether to round up (you cannot build 0.3 of a school).
• Is a headline sensible? If a news report says a state produced '5 crore tonnes of rice for each person', estimate: that cannot be right. Rounding and estimating protect you from wrong numbers.

The Go deeper layer works through census and election arithmetic in detail. Here is a project idea: pick one number from a newspaper each day for a week (crop output, railway passengers, cricket attendance, a state budget) and write one sensible calculation with it, with the operation and the answer's unit.

Try adding CXXIII + XLVIII in Roman numerals without converting to our digits. It is awkward: there are no columns, the symbols do not line up, and XL means 'ten before fifty', so the order of symbols changes their value. The answer is CLXXI (123 + 48 = 171), but getting there needs a lot of regrouping of symbols. Now try multiplying MCMLXXXVII by XLVII!

This is why the Indian place-value system with zero was such a revolution. With it, the same ten digits line up in columns, and the same short methods work for any size of number: carrying, borrowing, long multiplication and long division all depend on it. For centuries, Roman-numeral users did their actual calculating on counting boards or an abacus and only wrote the answers in Roman numerals.