# Problematics | Slide and rule

Aug 05, 2024 01:28 AM IST

## The 15 Puzzle is a familiar diversion where you slide tiles into a certain order. Here’s a version that requires you to slide same tiles into a different order.

In the 1880s, the legendary American puzzler Sam Loyd popularised the game we know today as the 15 Puzzle. Although Loyd tried to pass it off as his own creation (he had a dubious reputation in this respect; his English counterpart Henry Ernest Dudeney once accused him of stealing his puzzles and publishing them as his own), the game had actually been invented some years earlier by an American postmaster, Noyes Chapman.

Today, the game can be bought at stores and perhaps even off your local hawker. It consists of a 4x4 frame in which 15 numbered tiles are placed, with one square slot left blank. You need to slide the tiles around until you get the numbers 1 to 15 in order. It is not always easy, but it can be done.

The illustration above shows the final stages of a solver’s attempt to crack the puzzle. To bring the remaining few tiles in order, first move 12 into the gap above it, then slide 11 left into the new gap, then 15, then 12, then 11, and finally 15. Let this be the notation you use when you solve the puzzle that will follow: Simply name the tiles that you move in successive steps. In the above example, the moves would be notated as 12, 11, 15, 12, 11, 15.

#Puzzle 102.1

I got this puzzle from the writings of the late Russian mathematician Yakov Perelman, who attributes it to Sam Loyd.

## Starting from the position shown on the left, slide the tiles around until you reach the position on the right (and do remember to notate your moves in the manner described above).

#Puzzle 102.2

A cyclist pedals from one end of a road to another in 4 minutes 30 seconds. Turning and cycling back at the same speed, she returns to her starting point in 6 minutes. The reason she clocks two different times over the same distance, of course, is that a constant wind has been blowing, aiding her in one direction and deterring her in the other.

## If the wind stops blowing, how long will it take the cyclist to cover one leg of the journey?

MAILBOX: LAST WEEK’S SOLUTIONS

#Puzzle 101.1

Hi Kabir,

Let G, S and B denote the number of sportspersons who won only a gold, only a silver and only a bronze, respectively. Let the sportspersons who won two medals each be GS, SB and GB. Finally, let GSB be the number of sportspersons who won one medal of each kind and N be the number of persons who did not win any medal. This covers all the persons in the contingent.

From statement 2, GSB = 1.

From statement 3,

GS + G + S + N = 5

SB + S + B + N = 5

GB + G + B + N = 5

From statement 4,

GSB + GS = 2

GSB + SB = 4

GSB + GB = 3

Solving these equations, we get

GS = 1, SB = 3, GB = 2, B = 0, N = 1, S = 1, G = 2.

The size of the entire contingent is GSB + GS + SB + GB + G + S + B + N = 1 + 1 + 3 + 2 + 2 + 1 + 0 + 1 = 11.

— Professor Anshul Kumar, Delhi

Sampath Kumar V has sent a Venn diagram for the distribution of medals (#101.1). I have included it in the same illustration below as Sanjay Gupta’s visual solution to the geometry problem (#101.2).

#Puzzle 101.2

Dear Kabir

The answer is 60°. Drawing a similar line on the third surface of the die (blue) and removing the die from the picture leaves us with a perfect equilateral triangle. All angles will be 60° including the red-green.

— Sanjay Gupta, Delhi

Solved both puzzles: Professor Anshul Kumar (Delhi), Sampath Kumar V (Coimbatore), Sanjay Gupta (Delhi), Yadvendra Somra (Sonipat), Dr Vivek Jain (Baroda), Raghunathan Ravindranathan (Coimbatore), Harshit Arora (IIT Delhi)

Solved #Puzzle 101.1: Sabornee Jana (Mumbai), Ajay Ashok (Mumbai)

Solved #Puzzle 101.2: Dr Sunita Gupta (Delhi), Shri Ram Aggarwal (Delhi), Amarpreet (Delhi), Vivek Aggarwal (Bengaluru), YK Munjal (Delhi), Shishir Gupta (Indore)

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

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