Problematics | Russian eggs for sale

Shakuntala Devi is often described as more a calculating prodigy than a mathematician. Those who stress this do have a point, perhaps, given that she never had any formal training in mathematics, something that makes her genius all the more extraordinary. That said, let us not gloss over the fact that she was also very well-read. Her books of puzzles draw from several sources, sometimes uncredited, and sometimes difficult to locate.
One source Shakuntala Devi does credit is a 19th-century Russian poet called Benediktov, and the puzzle she describes apparently comes from the poet’s collection that was unpublished during his lifetime. I could not find any online reference to this puzzle from this poet.
Shakuntala Devi’s description of the puzzle comes across to me as incomplete. I am not sure if that was the way the Russian puzzle was originally designed or if it was Shakuntala Devi who omitted some instructions. The solution as described in her book, however, is well detailed. The description of the puzzle below is mine, with credit to the Indian genius as my source for the Russian puzzle.
#Puzzle 118.1
A mother gives 90 eggs to her three daughters: 10 to the eldest, 30 to the second daughter, and 50 to the youngest. She sends them to the market. “Sell them for a total that must not be less than 90 kopeks,” she instructs them. And then the catch: “Each of you must sell your eggs at the same price at any time, and each of you must bring home the same amount from your sales. I don’t want any competition within the family.”
Initially, the sisters are flummoxed by their mother’s instructions. How is it even possible to sell different numbers of eggs at the same price and still earn the same amount? Eventually, however, the sisters ingeniously work out a way:
1. At first, they divide their eggs into batches. One batch contains the same number of eggs for all three sisters. They decide this number after careful thought. They then sell each batch at the same price (same price per batch for every sister).
2. No sister’s stock of eggs (10, 30, or 50) is divisible into an exact number of batches. That is to say, after dividing the bulk of her stock into batches of the fixed size, each sister is left with a remainder of eggs that is less than one batch. This remainder may differ from sister to sister.
3. When all batches are sold and no more batches are possible, the sisters decide on a new price for the remaining eggs, to be sold singly. This price per single egg, again, is the same for all three sisters. The eggs are now costlier, but the sisters still manage to sell their remaining eggs at this new price.
4. In the end, each sister earns the same amount from her sales. And the grand total happens to be 90 kopeks, exactly the lower limit set by their mother.
#Puzzle 118.2

To repeat the rules of Wordle, the hidden word consists of 5 letters, which you seek to determine by using test words of 5 letters each. A green cell means this letter appears in the same position in the hidden word. Yellow means the letter appears in the hidden word, but in a different position. Grey means this letter does not appear in the hidden word at all.
MAILBOX: LAST WEEK’S SOLVERS
#Puzzle 117.1

Hi Kabir,
The two right-angled triangles, one above the square slab (shown with blue sides) and the other to its right (shown with red sides), are similar. Thus, if the height of the one above is h metres, the base of the one to the right would be 1/h metres. Now consider the large right-angled triangle with the ladder as its hypotenuse. Its height H = h + 1 and its base = 1 + 1/h. Therefore,
(h + 1)² + (1 + 1/h)² = 16
This equation can be solved to four possible values of h: (–0.203), (–4.920), (2.761) and (0.362). Discarding the negative values leaves two possible values of H, the height of the window from the ground, 3.761 m or 1.362. Correspondingly, B would be 1.362 or 3.761. Mathematically, both are valid solutions, although the first one appears consistent with the way the diagram is drawn.
— Professor Anshul Kumar, Delhi
Here I should mention Yadvendra Somra, who uses two different methods to solve the puzzle. One approach uses trigonometry, while the other approach leads to the same equation as in the solution from Prof Anshul Kumar above. Below is my summary of Yadvendra’s solution to the equation:
(1 + h)² + (1 + 1/h)² = 16
=> 1 + 2h + h² + 1 + 2/h + 1/h² = 16
=> h² + 2 + 1/h² + 2h + 2/h = 16
=> (h + 1/h)² + 2(h + 1/h) – 16 = 0
Yadvendra takes h + 1/h = z, reducing the above equation to
z² + 2z – 16 = 0
This had two roots. Rejecting the negative one, Yadvendra keeps z = √17 – 1, then substitutes this value in the original equation to solve another quadratic equation that gives h = 2.761 or 0.362. The height of the window follows.
#Puzzle 117.2
Hi Kabir,
The last digit in the three numbers was surely 5. So the answer should be 25 x 25 = 625.
— Dr Sunita Gupta, Delhi
 
Solved both puzzles: Professor Anshul Kumar (Delhi), Yadvendra Somra (Sonipat), Dr Sunita Gupta (Delhi), Sabornee Jana (Mumbai), Ajay Ashok (Delhi), Shishir Gupta (Indore), Shri Ram Aggarwal (Delhi)
Solved #Puzzle 117.2: YK Munjal (Delhi)