Problematics | Merry Xmas to all - Hindustan Times

Problematics | Merry Xmas to all

Dec 25, 2023 03:01 PM IST

The headline is a Christmas greeting but is also a hidden mathematical puzzle. Can you solve it, besides distributing some goods for the festive occasions?

This being the time of the year when most people celebrate, with some of them going on holiday, a couple of easier puzzles might suit the occasion. There will be some readers, of course, who believe that the best way to enjoy a festive occasion is to solve some really tough puzzles, but I will store those tough ones for later.

Welcome to Problematics!(Shutterstock)
Welcome to Problematics!(Shutterstock)

Both puzzles this week, therefore, are suited to the occasion. Not only in terms of easiness but also in that they are tailored to Christmas. The first of two Christmas puzzles coming up.

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#Puzzle 70.1

A barkeeper sometimes sells 750 ml bottles of various types of liquor, and sometimes pours them out in pegs for those who don’t need a full bottle. During the Christmas season, he soon runs out of full bottles. He is left with six partly-filled bottles. One of them has just had a small peg poured out of it, with 713ml remaining. The rest, in descending order, contain 460ml, 437ml, 414ml, 368ml and 345ml.

A customer drops in. “Give me all the liquor you have,” he pleads.

“No way,” says the barkeeper. “I need to keep some for other customers.” He sells the customer some of the liquor, without disturbing the contents of any of the bottles.

A second customer arrives. “Give me everything you have,” he echoes his predecessor.

“No way,” the seller repeats. “I need to celebrate too.”

So, he sells some of the liquor, again without disturbing the contents of any of the bottles.

He then picks up the unsold liquor. “This goes home with me,” he tells the customer. “Merry Christmas, and good night.”

He locks up and heads home. On the way, he observes that the first customer had bought exactly twice as much liquor (in terms of volume) as the second customer bought after him.

Which bottles went to which customer, and which did the seller take home?

#Puzzle 70.2

The second puzzle is actually tougher than the first, but it is not my own and has been around for a long time. It is, therefore, quite possible that you have across it before. If not, you may find yourself spending a little more time on it than you spent on the first puzzle.


In the greeting above, composed of 10 different letters, each letter stands for a different digit. The four numbers are to be read as you would read any number composed of digits placed in the units, tens, hundreds and thousands positions. For example, if we had the letters K, N, P and U, and if K was 1, N was 2, P was 3, and K was 4, then the word PUNK would have represented the number 3421.

To repeat, the digits 0-9 are all represented in MERRY XMAS TO ALL, and each letter stands for its own digit whenever it appears. Each of the four words, coincidentally, represents a number that is a perfect square.

What are those squares, and what digit does each letter represent?

Have a great season ahead. I have not yet made up my mind whether we should ring in the new year with easy or tough puzzles, but we shall definitely get back to more serious business in the week after that.


#Puzzle 69.1
#Puzzle 69.1

#Puzzle 69.1

Hi Kabir,

The following table illustrates the solution:

It can be solved in reverse. After G's distribution of money, all of them have 640 each. Likewise for F and so on. Below the highlighted diagonal above, the money should be doubled at each step, and above the diagonal it should be halved at each step.

— Sundarraj C, Bengaluru

#Puzzle 69.2

Hello Kabir Sir,

The possible combinations are:


Second word: MAGIC, GAMIC, MAFIC

— Abhishek Garg, Chandigarh

Thanks, Abhishek, for providing so many alternative words.

There was a mistake in the Wordle image when first published. The correct image, published later, ruled out the word STOLE. For fairness, however, I am counting STOLE as correct after it came from a couple of readers who might not have noticed that a change was made.

Solved both puzzles: Sundarraj C (Bengaluru), Abhishek Garg (Chandigarh), Yadvendra Somra (Sonipat), Professor Anshul Kumar (Delhi), Dr Sunita Gupta (Delhi), Ajay Ashok (Mumbai), Shishir Gupta (Indore)

Solved #Puzzle 69.1: Shri Ram Aggarwal (Delhi), Radhika Joshi (DPS Vasant Kunj), Kanwarjit Singh (Chief Commissioner of Income Tax, retired)

Solved #Puzzle 69.2: Anil Khanna (Ghaziabad)

Problematics will be back next week. Please send in your replies by Friday noon to

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    Puzzles Editor Kabir Firaque is the author of the weekly column Problematics. A journalist for three decades, he also writes about science and mathematics.

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