# Problematics | Mix and match in the chemistry lab

Aug 12, 2024 10:22 AM IST

## You have a mix of five chemicals which are not in the right proportions. How much of each must you add to get the proportions right?

Here is an announcement for dedicated puzzlers: a variety of online puzzles are now up daily in HT’s Games section. Many creators including myself are contributing crosswords of various kinds, while I am also setting a daily mathematical puzzle (which should be simple for Problematics readers) and a weekly Einstein puzzle. You will find all of it at hindustantimes.com/games.

Nothing changes in Problematics, of course. Here we are, with the latest pair of puzzles.

#Puzzle 103.1

“Do me a favour, please,” the chemist requests her mathematician husband. “Prepare a mixture in the proportions I have written down and do keep it ready for me when I return from class.”

“Sure,” says the husband, because the instructions look simple enough:

“Solution AUB#121: 18 drops; solution BVC#144: 24 drops; solution CWD#169: 27 drops; solution DXE#196: 30 drops; salt EYF#225: 18 grains. (PS: Use dropper FZG#256 for every solution. You need not worry about the salt; every grain in the bottle is exactly the same size.)”

Once the chemist has left, the mathematician enters his wife’s lab, his mind preoccupied with complex equations that are beyond the realm of Problematics. As a result, he gets all mixed up with the instructions. Luckily, he realises the error and makes a checklist as shown in the illustration.

“What needs to be done,” the mathematician tells himself, “is to add extra amounts of each component so that the mix ends up in the right proportions. The quantity will increase, of course.”

Being a mathematician, he wants to add the minimum necessary quantity of each chemical to bring the mixture to the specified proportions.

## How much of which should he add? (Hint: one of the five quantities remains unchanged.)

#Puzzle 103.2

To repeat what was mentioned last week, the game that we know as the 15 Puzzle was created by an American postmaster Noyes Chapman. The American puzzler Sam Loyd devised puzzles based on the game, including the puzzle that appeared last week as #Puzzle 102.1. Since we have had a puzzle from Loyd, it makes sense to follow it up with one from Henry Ernest Dudeney, Loyd’s English counterpart. Together, the two ruled the puzzling circles of their respective countries for several years.

In the following puzzle from Dudeney, I have changed the currency and the numbers to make the plausible.

You buy some pairs of shoes for a total cost of 60,000. You keep 15 pairs for yourself and sell the rest for a total of 54,000. This gives you a profit of 100 for each pair you have resold.

How many pairs did you buy and sell, and at what prices?

MAILBOX: LAST WEEK’S SOLVERS

#Puzzle 102.1

Dear Kabir,

The 15-tile puzzle kept me going. I actually went to a toy shop and bought the 15 Puzzle, and then tried solving it to bring it to your initial position. Then, I realized that the parity of the puzzles in the shop is not the same as the parity of the puzzle you had given. So, I had to tinker with the puzzle and change its parity, so that it can be brought to your initial position. Then, I tried solving the puzzle from your initial position to your final position. I was able to solve it, but it was so lengthy that I couldn’t capture the steps one by one. Rather I made a video of me solving your problem.

— Sampath Kumar V, Coimbatore

If you wish to view Sampath’s video of the solution, which he has put up on YouTube, please let me know and I shall mail you the link. For an illustrated version of the solution, the above is the painstaking effort from Professor Anshul Kumar.

#Puzzle 102.2

Dear Kabir,

Let us take distance =d metres, speed of cyclist= s m/min, speed of wind=w m/min

Time taken while cycling in the direction of the wind = d/(s + w) = 4.5 min

Time in the opposite direction = d/(s – w) = 6 min

Solving, we get d/s = 36/7 min

This is the time taken by the cyclist in either direction if there is no wind.

Solved both puzzles: Sampath Kumar V (Coimbatore), Professor Anshul Kumar (Delhi), Yadvendra Somra (Sonipat)

Solved #Puzzle 102.1: Sanjay Gupta (Delhi)

Solved #Puzzle 102.2: Dr Sunita Gupta (Delhi), Dr Vivek Jain (Baroda), Shri Ram Aggarwal (Delhi), Raghunathan Ravindranathan (Coimbatore), YK Munjal (Delhi), Harshit Arora (IIT Delhi), Sundarraj C (Bengaluru), Sanjay K Gupta

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

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