Once upon a time the Lord Abbott of St. Edmondsbury, in consequence of ’devotions too strong for his head,’ fell sick and was unable to leave his bed. As he lay awake, tossing his head restlessly from side to side, the attentive monks noticed that something was disturbing his mind; but nobody dared to ask what it might be, for the Abbott was of a stern disposition, and never would brook inquisitiveness. Suddenly he called for Father John, and that venerable monk was soon at the side of his bed. “Father John,” said the Abbott, “dost thou know that I came into this wicked world on a Christmas Even?” The monk nodded assent. “And have I not often told thee that, having been born on Christmas Even, I have no love for the things that are odd? Look there!” The Abbott pointed to the large dormitory window. The monk looked and was perplexed. “Dost thou not see that the sixty-four lights add up to an even number vertically and horizontally, but that all the diagonal lines are of a number that is odd except fourteen of them? Why is this?” “Of a truth, my Lord Abbott, it is of the very nature of things, and cannot be changed.” “Nay, but it shall be changed. I command thee that certain lights shall be closed this day, so that every line will have an even number of lights. See thou that this will be done without delay, lest the cellars be locked for a month and other grievous troubles befall thee.” Father John was at his wits’ end, but after consultation with one who was learned in strange mysteries (integer programming), a way was found to satisfy the whim of the Lord Abbott. Which lights were blocked up, so that those which remained added up to an even number in every line horizontally, vertically, and diagonally, while the least possible obstruction of light was caused? Father John held that the four corners should be darkened, but the sage explained that it was desired to obstruct no more light than was absolutely necessary, and he said, anticipating Lord Dundreary, “A single pane can no more be in line with itself than one bird can go into a corner and flock in solitude. The Abbott’s condition was that no diagonal lines should contain an odd number of lights.” (see  and ).
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!
Given a square board of size 4 × 4 cells. The game is to place tiles (pieces) on the cells. If a tile is placed on a cell then we say that the cell is “occupied” otherwise it is “vacant”. The goal is to place as many tiles on the square board as possible in a way that the sum of all vacant cells on the horizontal, vertical and the two diagonal lines for a occupied cell is exactly 6. (see  and ).
The first English puzzlist whose name has come down to us was a Yorkshireman – no other than Alcuin, Abbott of Canterbury (A.D. 735-804). Here is a little puzzle from his work, which is at least interesting on account of it’s antiquity. “If 100 bushels of corn were distributed among 100 people in such a manner that each man received three bushels, each woman two, and each child half a bushel, how many men, women, and children were there?” Now, there are six different answers, if we exclude a case where there would be no women. But let us just say that there were just five times as many women as men, then what is the correct solution? (see  and ).
Joshua is a biologist. His project for this term is measuring the effects of an increase in vitamin C in the diet of 9 laboratory rats. Each rat will have a different diet supplement of 1 to 20 units. Fractions of a unit are not possible. To get the maximum value for his experiment, Joshua has decided that for any group of three rats the supplements should not be in arithmetic progression. In other words, for three rats chosen at random, the middle supplement should be different from the arithmetic middle of the biggest and the smallest supplement. Thus, if the biggest supplement is 12 and the smallest is 6, for example, the middle supplement should not be 9 (9 being the arithmetic middle of 12 and 6). Find a set of supplements that Joshua could use (see  and ).
Five schools are camping on a public-school ground: Aldhouse, Bedminster, Chartry, Radford and Rugenham. The smallest contingent from the five schools was greater than 20 but less than 30. Aldhouse sent two less than half of the Rugenham contingent. The Radford and Rugenham contingents together were 14 greater than the combined Bedminster and Chartry contingents. The Bedminster and Rugenham contingents together were two less than half the total complement from the five schools while the Chartry and Radford contingents combined were 13/32 of that total. What was the strength of each contingent? (see  and ).
A butcher received an invoice for a consignment of 72 turkeys, but unfortunately it was smudged and a couple of figures were unreadable. All he could read was ’?67.9?’, with the first and last figures illegible. Nevertheless, being a “puzzler”, he was able to work out the price of a turkey immediately. What was the price of one turkey? (see ).
“Mrs Spooner called this morning,” said the honest grocer to his assistant. “She wants twenty pounds of tea at $2.85 per lb. Of course, we have a good tea $3.0 per 1b, a slightly inferior at $2.7, and a cheap tea at 2.1” “What do you propose to do?” asked the innocent assistant. “Do?” exclaimed the grocer. “Why, just mix up the three teas in different proportions so that the twenty pounds will work out fairly at the lady’s price. Just don’t put in more of the best tea than necessary, as we make less profit on that, and of course you will use only our complete pound packets. Don’t do any weighing.” How was the poor fellow able to mix the three teas? Could you have shown him how to do it? (see  and ).
A man who possesses a half-sovereign, a florin and a sixpence goes into a shop and buys goods worth 7 shillings and 3 pence. But the shopkeeper cannot give him the correct change, as his coins are a crown, a shilling, and a penny. Luckily, a friend comes in the shop, and finds that he has a double-florin, a half-crown, a fourpenny piece and a threepenny bit. Can the shopkeeper effect an exchange that will enable him to give the man the correct change, and to give his friend the exact equivalent of his coins?
Tommy was given 15 coins for his birthday (half-crowns, shillings and sixpence). When he added it up, he found that he had £1. 5s. 6d (one pound 5 shillings and 6 pences, see below). How many half-crowns was he given?
A farmer leaves 45 casks of wine, of which 9 each are full, three-quarters full, half full, one quarter full and empty. His five nephews want to divide the wine and the casks, without pouring wine from cask to cask, in such a way that each receives the same amount of wine and the same number of casks, and further so that each receives at least one of each kind of casks, and not two of them receive the same number of every kind of casks (see  and ).