Problematics | The mathemagic of dice

The term ‘mathemagic’, which we use so frequently in puzzle literature, is said to have been coined by Royal Vale Heath (1883–1960), a New York stockbroker, writer, magician and puzzler. ‘Mathemagic’ is the title of a book Heath published in 1933.

Among the mathematical games Heath created, one was called the ‘Di-Ciphering Trick’ as described by the late mathematics writer Martin Gardner, who was greatly influenced by Heath’s work. The game is now marketed in a woodcraft edition under the title ‘Heath’s Deciphering Dice’ by Creative Crafthouse, an American company. I found that it is on sale online (see image), but the description below is based on Gardner’s writings.

#Puzzle 92.1

The game consists of five dice, with the six faces inscribed with numbers rather than the spots we have on ludo dice. The numbering is as follows:

Die #1: 483, 285, 780, 186, 384, 681

Die #2: 642, 147, 840, 741, 543, 345

Die #3: 558, 855, 657, 459, 954, 756

Die #4: 168, 663, 960, 366, 564, 267

Die #5: 971, 377, 179, 872, 773, 278

The game (or rather, the trick) is played with the magician asking a spectator to roll the five dice. The magician then looks at the dice and announces the total of the numbers on the five top faces. He gives the sum in an instant, creating the impression of possessing magical powers.

The trick is, of course, mathematical. The magician simply takes the last digit from each of the five numbers on the dice tops, then carries out a simple arithmetic exercise to obtain their total. He does not actually need to add up the five three-digit numbers. Indeed, there are many people who can add that many numbers in very quick time, but let us focus solely on the short cut that the magician uses.

What is the magician’s short cut, and what is the mathematical principle that makes this method work?

#Puzzle 92.2

A man is half as much again (that means 1.5 times) the age of his wife, who, in turn, is 6 times as old as their son, who is twice as old as his sister. The man’s mother’s age is the sum of the ages of the other four mentioned.

Within credible limits, how old is each member of the family (integers only)?

MAILBOX: LAST WEEK’S ANSWERS

#Puzzle 91.1

Hi Kabir,

The solution to #Puzzle 91.1 is as shown in the illustration.

— Raghunathan Ravindranathan, Coimbatore

#Puzzle 91.2

Hi Kabir,

At first look, it would appear that option 1 with one annual increment of ₹1.2 lakh is a clear winner over option 2 with two increments of ₹30,000 in a year. However, a closer consideration reveals just the opposite. Let us see how the annual salary would change from year to year for each option. For option 1, obviously, the annual salary would go up by ₹1.2 lakh per year. For option 2, the salary received in the first half of a year would be two increments of ₹30,000 (= ₹60,000) more than that received in the first half of the previous year. Similarly, the salary received in the second half of a year would be two increments of ₹30,000 (= ₹60,000) more than that received in the second half of the previous year. Thus, there would be a total increase of Rs. 1.2 lakh with respect to the previous year. With the annual increase being equal in both cases, it is the annual salary in the first year that becomes the deciding factor. The annual salary in the first year is ₹6 lakh in option 1 and ₹6.3 lakh (3 lakh + 3 lakh + 30,000) in the second option. Therefore, option 2 is undoubtedly better. The table makes a precise comparison of the two options (all figures are in lakh rupees).

— Professor Anshul Kumar, Delhi

​***

Hi Kabir,

We can see that in the second option, the total yearly earnings are ₹30,000 more every year than in the first option.

— Shishir Gupta, Indore

Solved both puzzles: Raghunathan Ravindranathan (Coimbatore), Prof Anshul Kumar (Delhi), Shishir Gupta (Indore), Anil Khanna (Ghaziabad) Dr Sunita Gupta (Delhi), Sampath Kumar V (Coimbatore), Rituparna Gupta (Indore), Ajay Ashok (Mumbai), Sanjay S (Coimbatore), Shruti M Sethi (Ludhiana)

Solved #Puzzle 91.1: Dr Anjali Kashikar (Mumbai), Geetha G (Cuddalore), Amardeep (Delhi), Akshay Bakhai (Mumbai), Dr Vivek Jain (Baroda), Yadvendra Somra (Sonipat)

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