Problematics | A piece of cake

One of five girls slyly eats a slice of cake their teacher had kept for herself. Which one?

Thank Heaven for Little Girls, as Maurice Chevalier sang in the 1958 musical Gigi, a landmark film that won multiple Oscars. Little girls are the subject of our logic puzzle today. Also, one might add, Thank Heaven for Liars. For, if everyone spoke the truth all the time, we wouldn’t have had puzzles of this kind.

A piece of cake (PIXABAY)

#Puzzle 208.1

Teacher takes her class of 23 girls on a picnic. While the girls are playing, she cuts a nice round cake into 24 equal slices. The 24th slice, of course, is intended for Teacher herself. Teacher then walks up to the girls, who are in various groups, and calls them over. When she distributes the slices, however, the teacher finds that there are only 23. The slice she had kept for herself was eaten by one of the girls. The culprit must be one of the five girls whom she had called over in the first batch. When the teacher grills them, here are their responses.

Anjana: I did not eat the missing slice

Anubha: I did not eat the missing slice

Ananya: I did not eat the missing slice

Anjali: I did not eat the missing slice

Ankita: I did not eat the missing slice

Anjana: Anubha ate the missing slice

Anubha: Anjana is falsely blaming me

Ananya: Anjana did not eat the slice

Anjali: Ankita and I had coffee together

Ankita: I never drink coffee

Anjana: I don’t like cake

Anubha: Ankita ate the slice

Ananya: Anubha is falsely blaming Ankita

Anjali: Anubha ate the slice

Ankita: Ananya ate the slice

Later, the teacher finds out who ate the slice. She then reflects on the 15 statements from the five girls, and observes that each girl gave one false and two true statements.

Who ate the slice, and which five statements (one from each girl) are false?

#Puzzle 208.2

Two cans containing coffee are the same shape and made of the same material. One is 12 cm high and weighs 2.1 kg, while the other is 6 cm high and weighs 275 grams.

What is the net weight of the coffee in each can?

MAILBOX: LAST WEEK’S SOLVERS

#Puzzle 207.1

Last week’s cows-and-meadow puzzle contained a little ambiguity. The cows started eating on Day 0 and finished on Day 15, which can mean 15 days if you take that to mean the start of Day 15 (the intended meaning) or 16 days if both Days 0 and 15 are included. Some readers have chosen 15 days, and some 16.

Dear Mr Kabir,

I liked this puzzle. I also remember a similar one that was printed in the Monday edition of the newspaper a long time ago. Here is the solution.

Let the initial amount of grass be R and, let the daily growth rate of the grass be r. Also, let the amount of grass eaten by each cow daily be D.

Case 1: 20 cows can finish the grass in 24 days.

Gross amount of grass in 24 days = Amount eaten by the cows in 24 days

R + 24r = 20 x D x 24 = 480D

Case 2: After 20 cows have eaten for 6 days, 25 cows can clear the field in 12 more days.

Grass remaining after 6 days = (R + 6r) – (20 x 6 x D) = R + 6r – 120D

New grass grown in 12 days = 12r

Grass eaten by 25 cows in 12 days = 25 x 12 x D = 300D

(R + 6r – 120D) + 12r = 300D

=> R + 18r = 420D

Subtracting the 24-day equation (R + 24r = 480D) from the above (R + 18r = 420D), we get

6r = 60D, or r = 10D

Using this value also gives us R = 24r. This is the initial amount of grass.

Case 3: 20 cows for 6 days, 25 cows for 4 days, and (25 + x) cows for 5 days, where x is the number of new cows added after 10 days

Grass remaining after 6 days = R + 6r – 120D = 24r + 6r – 12r = 18r

New grass grown in the next 4 days = 4r

Grass eaten by 25 cows in 4 days = 25 x 4 x D = 100D = 10r

Grass remaining after 10 days = 18r + 4r – 10r = 12r

After x cows are added,

Grass grown in next 5 days = 5r

Total grass in the next 5 days = 12r + 5r = 17r

Grass eaten by (25 + x) cows in 5 days = (25 + x) x 5 x D = (25 + x)(r/2)

Equating,

17r = (25 + x)(r/2)

After cancelling out r and solving,

x = 9

So, 9 more cows are added on Day 10.

— Shri Ram Aggarwal, Palam, New Delhi

***

Hi Kabir,

Let C denote the amount of grass consumed by a cow in a day and let G denote the amount of growth of grass in a day. The farmer goes through 3 different plans. We can estimate the initial amount of grass in the meadow by these three plans as follows.

Original plan: 20 cows eating grass for 24 days minus grass growth for 24 days = 480C – 24G

Second plan: 20 cows eating grass for 6 days plus 25 cows eating grass for 12 days minus grass growth for 18 days = 420C – 18G.

Final plan: 20 cows eating grass for 6 days plus 25 cows eating grass for 4 days plus (25+N) cows eating grass for 6 days minus grass growth for 16 days = (370 + 6N)C V 16G, where N is the number of cows added on Day 10.

This gives 480C – 24G = 420C – 18G = (370 + 6N)C – 16G.

Adding 16G – 370C to each side and dividing throughout by 2 gives

55C – 4G = 25C – G = 3NC

From these we get G = 10C and N = 5.

The number of cows added on Day 10 = 5.

This is provided the job is finished in 16 days from Day 0 to Day 15. If the total period is taken as 15 days, then the number of cows added on Day 10 is 9.

— Professor Anshul Kumar, Delhi

#Puzzle 207.2

Hi Kabir,

The ₹1500 should be divided in the ratio 1:2, i.e. ₹500 to the older man and ₹1000 to the younger man

For the older man, let y be the rate of removal of books and x be the rate of arrangement. Therefore, for the younger man, the rates are 4x (removal) and y (arrangement). It is given that tthe ratio between the two jobs is equal for each man.

y/x = 4x/y

=> y²= 4x²

=> y = 2x

If we take x = 1, the younger man removes 4 books in 1 unit of time, then arranges these 4 books in 2 units of time. On the other hand, the older man would remove 2 books in 1 unit of time, and arrange these 2 books in 2 units of time. Thus, in a given span of time, the younger man does twice as much work as the older man, and hence should receive twice as much money.

— Sabornee Jana, Mumbai

Solved both puzzles: Shri Ram Aggarwal (Delhi), Professor Anshul Kumar (Delhi), Sabornee Jana (Mumbai), Vinod Mahajan (Delhi), Yadvendra Somra (Sonipat), Dr Sunita Gupta (Delhi), Shishir Gupta (Indore), Kanwarjit Singh (Chief Commissioner of Income-tax, retired), Ajay Ashok (Delhi)

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