Problematics | Board games with 3 dice and chess pieces

Sep 11, 2023 12:45 PM IST

Here's party trick first described in 1612, played with dice. Based on how the spots ate configured on standard dice, can you figure out why the trick works?

Dice, like all other tools of gambling and indoor games dictated by chance, can be the source of many puzzles. Not all these puzzles, however, are based on probability. Some of them derive from the standardised arrangement of the spots on the six faces.

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

In a “standard” die, the face with 1 spot is opposite the face with 6 spots, 5 spots are opposite 2 spots, and 4 are opposite 3. That is the way it should be, although you may occasionally come across locally-made dice that do not follow this arrangement.

#Puzzle 55.1
#Puzzle 55.1

Remember the smart alec you meet at every party? He’s back, with another trick to play on you. He hands you three dice, in which the spots are in the standardised configuration described above. He then turns his back, and asks you to roll all three dice on the table. You roll, after which he gives you a series of instructions:

Step 1: “Add the spots on the tops of the three dice, but don’t tell me the total.”

Step 2: “Pick up any two, look at their bottom faces, and add those spots to the total from step 1.”

Step 3: “Roll those two dice again. Then, add the spots on their top faces to the existing total.”

Step 4: “Pick up any one of the two, and add the spots on its bottom face to the total.”

Step 5: “Roll this single die, which is singular for dice,” the smart alec adds, as you didn’t know. “When the die stops,” he continues, “add the spots on the top face to the existing total.”

Once you’re done, the smart alec turns around, looks at the dice, and announces the total you have calculated.

How does the smart alec work out the total?

(Adapted from Mathematical Magic Show by Martin Gardner, who attributes the trick to a book from 1612.)

#Puzzle 55.2
#Puzzle 55.2

The bishop, as every chess player knows, moves diagonally across the board. What is the highest number of bishops that you can place on a chessboard so that none of them is in a position to capture any other bishop? Send illustrations, please.

Mailbox: Last Week’s Solvers

#Puzzle 54.1

Hi Kabir,

When running in the same direction, it takes 6 minutes and 30 seconds for the two runners to meet again. This means it takes 6:30 minutes for the coach to run 390 m more than the novice. This works out to 1 metre per second, which is the difference between their respective speeds.

When running in opposite directions, it takes 30 seconds to cover 390 metres. This works out to 13 metres per second, which is the sum of their respective speeds.

From the above, we get individual speeds of 6 m/s and 7 m/s. To complete a 400 m run, the coach will take a little over 57.1 seconds and the novice will take a little over 66.6 seconds.

— Akshay Bakhai, Mumbai

The Sun is 10¹⁶ times bigger than the bacterium. Let the bacterium, doubling every day, take x days to grow to this size.

2ˣ = 10¹⁶,

Or, x = 16/log 2 = 53.15

At the end of 53 days, the bacterium will be slightly smaller than the Sun, but after 54 days it will be bigger. So, after 53 days, it needs to push the STOP button.

— Harshit Arora, IIT Delhi

Solved both puzzles: Akshay Bakhai, Harshit Arora, Bhasker Mundhra, Dr Sunita Gupta, Sunita & Naresh Dhillon, Rajesh Bansal, Kanwarjit Singh, Ranjan Ghosh, Col (Dr) Sudhakar Tyagi, Gp Capt RK Shrivastava (retd), Harsh Ozare, Shri Ram Aggarwal, Vinod Mahajan, Shishir Gupta, Sabornee Jana, YK Munjal, Aakashneel Saha, Aparajita Shrivastava, Prof Anshul Kumar, Priye Rana, Rajeev Chauhan, Vishesh Sethi, Amardeep Singh

Solved #Puzzle 54.1: SC Vasudeva, Dr GL Arora, Prakash Bhate, Yadvendra Somra, Anil Khanna, Jatinder Singh Gill, Asha Nanda, Anubhuti Singh, Vikas Nanda, Mehak Gupta, Amar Lal Miglani, Amit Kumar Gupta, Sumit Malhotra, Bimal D Jhaveri, Ajay Ashok, Megha Singh, BS Ramachandra Sai, Amit Gupta, Soumil Mukhopadhyay, Mudit Singhal, Narendra Prasad

Solved #Puzzle 54.2: Ananyaa Priyadarshini, Ananya Arvind, Nikhil Yadav, Charvi Brajpuriya, Khushboo Yadav, Abhishek Garg, Dr Nakul Makkar, V Anand, Jawahar Lal Aggarwal, Surinder Seth, Narendar Kumar Aggarwal, Raunaq Nayar

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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