## Remainder Puzzle II (math04a)

This problem is the answer to the fourth question of model math04: Find the 10 smallest numbers with the remainders of 1, 2, 3, and 4 when divided by 3, 4, 5, and 6.

In this blog, I will add a new post all 3-5 days. Each post contains a complete (optimization) problem, many of them are puzzles. Each problem consists of a comprehensive description of the problem, a description of the modeling (translation from spoken language to mathematical language), a mathematical formulation, an implementation in the LPL modeling language, and a link to solve the problem directly through the Internet. A full document is also downloadable as a PDF.

I start with easy puzzle problems, but later on, more complicated problems will be presented. Enjoy!

This problem is the answer to the fourth question of model math04: Find the 10 smallest numbers with the remainders of 1, 2, 3, and 4 when divided by 3, 4, 5, and 6.

Is there a number which gives a remainder of 1 when divided by 3, and a remainder of 2 when divided by 4, and a remainder of 3 when divided by 5 and finally a remainder of 4 when divided by 6? Find the smallest such number, if any exists. (see [1] and [2]).

A woman has been carrying a basket of eggs to the market when a passer-by bumped into her. She dropped the basket and all the eggs were broken. The passer-by, wishing to pay for her loss, asked, “How many eggs were in your basket?” “I don’t remember exactly”, the woman replied, “but I do recall that when I divided the number of eggs by 2, 3, 4, 5 or 6 there was always one egg left over. When I took the eggs out in groups of seven, I emptied the basket.” What is the smallest possible number of eggs that broke? (see [1] and [2]).

The castle of Lord Hamilton is threatened from all four sides. Therefore, he places 12 guards on the top of its highest tower to observe the surroundings day and night. The guards can be placed on 12 platforms (the positions are numbered from 1 to 12 in Figure 1) – at each side there are 4 positions. Guards on a side platform can look only in one of the four directions, guards on the corner platforms 1, 4, 9, and 12 can look at two sides. Hence, the corner platform belongs to two sides.

Eva recently discovered a shelf full of bottles of wine in the cellar. She counted 24 bottles of Rioja and 17 bottles of Malbec. She also noticed the price labels on the bottles: The price of one bottle of Rioja was $25 and one bottle of Malbec costs $49. Suddenly, she realized that her husband had spent $2000 a week ago for just this wine. How many bottles did the husband already drink? (Let x and y be the number of bottles of Rioja and Malbec bought by the husband.)