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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!

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Coin Puzzle I (math06)

Suppose that 11 coins with round holes are worth 15 drachms, while 11 coins with square holes are worth 16 drachms, and 11 coins with triangular holes are worth 17 drachms. How many round, square or triangular pieces of cash would be required to purchase an item worth 11 drachms, using the least number of coins? (see [3] and [1]).

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The Mangoes Puzzle (math05)

Three men who had a monkey bought a pile of mangoes. At night one of the men came to the pile of mangoes while the others were asleep. He observed that removing one of the mangoes would leaves an amount divisible by three. He then tossed the extra mango to the monkey and took away one third of the remainder. Finally, he went back to sleep. Presently, another one of them woke up and went to the pile of mangoes. He also found just one too much to be divided by three so he tossed the extra one to the monkey, took one third of the remainder, and returned to sleep. After a while the third rose, and he too gave one mango to the monkey and took away the whole number of mangoes which represented precisely one third of the rest. The next morning the men got up and went to the pile. Again they found just one too much, so they gave one to the monkey and divided the rest evenly. What is the least number of mangoes with which this can be done? (see [2] and [1]).

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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.

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Remainder Puzzle I (math04)

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]).

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Egg Basket Puzzle (math03)

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]).

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Twelve Guards (math02)

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.

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Drunken Husband (math01)

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.)

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