The Bendiktov Problem

           The Great Russian poet Benediktov (1807 – 1873) was very fond of mathematics and he collected and compiled a whole volume of tricky brain teasers.

            Though his work was never published, the manuscript was found in 1924. An interesting problem contained in the manuscript, captioned ‘An Ingenious Way of solving a tricky Problem’ goes as follows:

One woman who made a living by selling eggs had 90 eggs which she wanted to sell, so she sent her three daughters to the market giving 10 eggs to her eldest and cleverest daughter, 30 to the second and 50 to the third.

            ‘You’d better agree among yourselves,’ she told them, ‘about the price you’re going to ask for the eggs, and keep to it. Stick to the price you decide upon. But I hope that, in spite of your agreement, the eldest  ,being the cleverest will receive as much fir her ten eggs as the second will receive for her thirty and will teach the second to sell her fifty. In other words, each of you is to bring home the same amount, and the total for the 90 eggs is not to less than 90 kopeks.’

              A kopek may be treated as a rupee. How do you think the girls carried out the instructions of their mother?

Answer:    Thus goes the Benediktov story.’ The problem was a tricky one and the three daughters discussed it on ‘their way to the market, with the second and third appealing to the eldest for advice.

The latter thought for a while and then said:

‘Look sisters, we’ll sell our eggs even at a time and not ten as we always do. We’ll fix a price for seven eggs and stick to it, as mother has told us to. Mind you, don’t reduce the price however much people may bargain! We’ll ask three kopeks for the first seven eggs, all right?’

‘That’s pretty cheap!’ the second sister interjected.

‘Never mind,’ the eldest retorted,’ we’ll raise the price for the eggs that remain after that. I have made sure that there won’t be any other egg vendors at the market. So there’ll be no one to force our prices down. And when there’s demand for eggs and not many of them are left, the price goes, up, that’s only natural

And that’s exactly where we’ll make up’.

‘And how much shall we ask for the remaining eggs? The youngest sister asked.

‘Nine kopeks and egg. And believe me; people who need eggs will pay the price.’

‘That‘s pretty stiff,’ the second sister remarked.’so what? The first seven – egg batches will be cheap. The expensive eggs will make up for the loss.’

‘The sisters agreed.

          ‘At the market each chose a place. The cheap price brought on an avalanche of buyers and the youngest, which had 50 eggs, soon sold all her eggs but one. At three kopeks per seven eggs she made 21 kopeks. The second sister, who had 30 eggs, made 12 kopeks by selling four people seven eggs each, and had two eggs left in the basket. The eldest made 3 kopeks from the sale of seven eggs and was left with three eggs.

          ‘Suddenly a cook appeared with instructions to buy ten eggs. Her mistress’s sons had come home on leave and they loved omelets. The cook rushed about the market, but the only six eggs- the youngest had one, the second two and the eldest three.

            ‘It is only natural that the cook rushed to the one who had three- that is, to the eldest sister who had sold her batch of seven eggs for 3 kopeks.

‘How much d’you want for your eggs?’ she asked.’nine kopeks and egg, ‘was the reply.

‘What them or leave them, these are my last and I won’t take a kopek less.’

‘The cook ran to the second sister, the one who had two eggs left in her basket.

‘How much?’ she yelled.

‘Nine kopeks and egg. That’s the price and these are my last eggs.’

‘And how much do you want for your egg?’ the cook turned to the youngest sister.

‘Nine kopeks.’

‘Well, there was nothing the cook could do, so she bought the eggs at this exorbitant price.

‘All right,’she burst out,’ I’ll take the lot.’

              ‘She paid 27 kopeks to the eldest sister for her three eggs and with the three kopeks the latter had from the earlier sale this brought her total receipts to 30 kopeks. The second sister got 18 kopeks and with the 12 kopeks she had received earlier that also made 30 kopeks. The youngest got 9 kopeks for the remaining egg and that, added to the 21 kopeks she had made on the sale of 49 eggs, brought the total also to 30 kopeks.

           ‘The three sisters then returned home, gave the money to their mother and told her how, sticking to the price they had agreed upon , they had succeeded in selling ten eggs for the same price as 50.

‘Their mother was very pleased that her instructions had been carried out and that her eldest daughter had proved so clever. But she was even happier that her daughters had brought her exactly what she had told them to bring – 90 kopeks.