December 10, 2023

By day, Shyam Sunder Gupta was Principal Chief Engineer of Indian Railways. By night, he was a guru of recreational mathematics.

For decades, Gupta spent his free time exploring patterns in numbers, his numerical curiosities finding their way into journals, magazines and books.

Now, recently retired from the railways, he has published a book of puzzles, from which today’s five number conundrums are taken.

1. Brahmagupta’s basket

The Indian mathematician Brahmagupta during the 7th century AD posed the following problem:

When eggs in a basket are taken out 2, 3, 4, 5 and 6 at a time, there remain 1, 2, 3, 4 and 5 eggs respectively. When they are taken out 7 at a time, none are leftover.

Find the smallest number of eggs that could be in the basket.

2. The biggest number

If the average (the mean) of 20 different positive whole numbers is 20, find the largest possible value that any one of the numbers can have.

3. Nine numbers

Find the missing numbers X and Y in the following list:

1100100, 10201, 1210, 400, 244, 202, X, Y, 100

4. Match squares


Imagine this image is of 40 match sticks arranged to make a 4×4 square.

A total of 30 squares (1 of 4×4, 4 of 3×3, 9 of 2 x2 and 16 of 1×1) can be seen in this arrangement. Find the minimum number of match sticks which on removing, vanishes all 30 squares. (i.e. the perimeter of all of them is broken.)

5. Square sums

Arrange the numbers from 1 to 15 in a row such that the sum of every two adjacent numbers is a square number (i.e. 1= 12, or 4= 22, or 9 = 32, etc).

I’ll be back at 5pm UK with the solutions. PLEASE NO SPOILERS.

UPDATE: The solutions can be read here.

Please talk about trains, or the amazing history of ancient Indian mathematics.

Shyam Sunder Gupta began his love affair with numbers in the 1970s when he was a student. His investigations into the properties of numbers are legendary in recreational maths circles. He is, for example, the second most prolific contributor to the Prime Curios website, which lists interesting things about prime numbers. Many of his discoveries are on his own website

“My love of recreational maths not only keeps me stress-free but also stimulates creative thinking and helps me solve many problems,” he says.

Today’s puzzles all come from his excellent new book, Creative Puzzles to Ignite Your Mind. I selected some of the simpler puzzles – if you want more challenging problems, the book has lots of great material.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

I give school talks about maths and puzzles (online and in person). If your school is interested please get in touch.

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