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Liu

Ziyuan Liu

Math 07

5/16/2013

Historical Math Puzzles

Puzzles from the ancient times to present day present the curiosities of the mind and the seed for innovation. From ancient Egyptian papyrus, which shows estimations of pi, to the legendary grains on a chessboard, historical math puzzles provide a shoulder modern mathematics. Not only do these puzzle provide a great story, but most also provide solutions or outcomes that may surprise many common readers.

Motivation and Significance

I am convinced that historical puzzles are, indeed, one of the best puzzle genres because of their ability to amaze and theircloseness to application. Many puzzles originate from the need for innovation in society–almost all historical puzzles contain some form of application to academia. However, math puzzles are not just created for the puzzles or application.A recent discovery showed that the 3,600 year old Rhind papyrus scroll, which estimates pi, alsocontains a puzzle similar to the St. Ive’s riddle:

“As I was going to St Ives

I met a man with seven wives

Every wife had seven sacks

Every sack had seven cats

Every cat had seven kits

Kits, cats, sacks, wives

How many were going to St Ives?”[1]

This puzzle is designed as a nursery rhyme for learning counting and multiplication. However, on the ancient Egyptian papyrus, the only difference is the background story and the object names [2]. The sum total of all things mentioned in both versions turns out to be 19,607–keep in mind that both versions are introduced three millenniums apart. Not only did this puzzle prove the importance of historical puzzles, but it also demonstrates that recreational mathematics is not a novel activity.

Example 1 – Grains on a chessboard

A favorite of puzzle of mine has always been the wheat on the chessboard game. The puzzle itself is derived from a legend. Around the 10th century CE, an Indian King was so ecstatic with the invention of chess that he promised the inventor the right to name any reward. The clever inventor quickly made up his mind and requested a single grain of wheat on the first square of the board, two grains on the second, four grains on the fourth, and so on.

Figure 1. The calculation of number of grains up on the chessboard.[1]

The unlucky King, ignorant of exponential growth, surprisingly agreed to this proposal. As one can see from Figure 1 (where K stands for thousand, M for million, and G for billion), the number increases exponentially from one square to the next. A romanticized version of the tale speaks of the King eventually giving up his throne to the inventor of chess because the total worth of all the grains combined outweighed his whole empire.

The general importance of this puzzle is taught through the moral of the tale: do not underestimate the value of geometric growth. In this puzzle, the total number of grains to be awarded can be computed from the following:

The puzzle revolves around the growth due to the summation of 2 to the power of each square’s number. If the equation is expanded, the value of each subsequent term is doubled. For example, the 32nd term would be: 2^31 = 2147483648, but the 33rd term would be: 2^32 = 4294967296. The sheer quantity of grains of the last square alone accounts for more than half of the total quantity. In this prospective, it is not hard to see why the foolish king now owes the inventor of chess enough wheat to buy multiple kingdoms.

Example 2 – Eight Queens

In continuance with chessboard puzzles,Max Bezzel proposed another one in the 19th century. Bezzel, a chess enthusiast, pondered the following placement of chess pieces: eight queens on an eight by eightchessboard with each in a non-losing position. This puzzle was proved to be very computationally expensive. By hand, it took over two years for a general solution to be provided for n queens in ann-by-n chessboard.

Figure 2. One of the solutions to the eight queens problem.

Figure 2 shows just one of twelve unique solutions to the eight queens puzzle on an eight-by-eight chessboard. Being computationally expensive for people back in the 19th century, this puzzle was also considered to be very computationally expensive for computers in the 1980s. However, unlike the previous puzzle, this puzzle prompted computer scientists to recreate a rather intelligent method called backtracking to efficiently solve such a puzzle. Originally, a computer would put 8 queens on any 8 opened spaces on the board and check for the validity of such a move. However, this method would invoke 64 choose eight possibilities, where out of 64 squares, eight can be placed anywhere. However, most of the combinations were invalid since not all pieces were in valid positions. Backtracking allows a computer to check each move for validity. For this puzzle and the sake of obviousness, another rule is imposed: each piece must occupy its own column. Therefore, with backtracking, the computer checks the validity of the board column by column until the condition is fulfilled. If the piece were in the wrong position, the program, like an intuitive human being, would retract that move and put the piece down in the next available square. This puzzle allowed us to gain insight on early artificial intelligence where the computer reasons and corrects moves.

Example 3 – Emperor’s gold

My father introduced this next puzzle to me recently; however, like the first puzzle, it wavers in historical accuracy. A Han emperor demanded tributes from his ten states. The states, neither wanting to spend more nor be surpassed by any other states, each decided to send in a ten-kilogram block of gold (which, for the sake of imagery is about 10 palm-sized gold bars. See figure 3). The emperor was quickly told by one of his court spies that one of the ten states decided to cheat, and infused a nine-kilogram block of gold with one-kilogram bronze or copper. The problem was that all the goldsent in from each of the nine states were somehow indistinguishable. Furious from such trickery, the emperor gave his unfortunate treasurer the task to figure out which block of gold is not pure. Feeling extremely cynical, the emperor allowed his treasurer only one use of the measuring scale, which should be enough to distinguish the faux gold; if the treasurer is unable to do so with one try, he will be executed.

Figure 3. A kilogram bar of gold.

Luckily, I was not the treasurer, as I did not figure out this puzzle until Google led me to other versions of the same puzzle. Since the gold blocks are indistinguishable appearance-wise, the trick to this puzzle is to cut an incrementing amount from each gold block to measure. For the purpose of simplicity, let the density of gold be 20 grams per cubic centimeter and the density of copper (what we will use in this case) be 10 grams per cubic centimeter–so let the difference in density be exactly 10 grams per cubic centimeter. Thus, the density of one cubic of impure gold is (20(9)+10(1))/10 = 19 grams per cubic centimeter. Now, the first step is to cut exactly one cubic centimeter from the first block of gold, two cubic centimeters from the second block, three from the third, and so on and so forth. Now, if the ten pieces of newly cut gold were truly one hundred percent gold, the total weight would be (1+2+3+4…+10)*20 or 55*20 =1100 grams. However, since we know there is exactly one piece of impure gold, we will weigh all ten newly cut pieces to examine the difference in the actual weight from the theoretical weight. If the actual weight was 1099 grams, we be can be sure that the impure gold was in the one cubic centimeter piece, because the difference in density between pure gold and the impure gold is one grams per cubic centimeter. Thus, if one obtains weightsfrom 1090 to 1099, one can easily distinguish the impure gold due to the different size of the gold pieces. This puzzle is related to many weighing puzzles where the trick is to make the indistinguishable pieces distinguishable through manipulation.

Analysis and discussion

These three puzzles, indeed, provide very interesting stories to the common reader. The grain on the chessboard introduces the scale of exponential growth, where a seemingly small and insignificant quantity can, without control, grow to a massive quantity.The eight queens puzzle presents a very easy and straightforward problem to do, however, the problem gets exhaustive if one is asked to find all of the possibleplacements of the queens. Introducing the backtracking technique, computer scientists showed that this kind of repetitive problem is most efficiently solved with a computer. Lastly, the emperor’s gold, unlike the first two puzzles, requires more thinking outside-of-the-box and intuition. However, it should be noted that for simplicity we assumed that the volume of 1kg of copper is the same as 1kg of gold when it would make more sense to mix half-kilogram of copper with nine-kilograms of gold to maintain the same volume. Regardless, with the same cutting method, the treasurer can easily find the faux gold without looking at the overall volume.

Conclusion

Historical puzzles, as demonstrated, bring much context in relation to either modern culture or innovation. These types of puzzles are truly my favorite genre due to the general ease of comprehension and the lack of cultural or prerequisite knowledge needed. These puzzles not only provide a retrospective view on curiosities that were indulged by recreational mathematics, but also great methods to solving novel problems.

Works Cited

[1] Opie and P. Opie, The Oxford Dictionary of Nursery Rhymes (Oxford University Press, 1951, 2nd edn., 1997), pp. 376-7.

[2]Belluck, Pam. "Math Puzzles' Oldest Ancestors Took Form on Egyptian Papyrus."The New York Times. The New York Times, 07 Dec. 2010. Web. 15 May 2013.

[3] Art. XIII.—The Origin and Early History of Chess, A. A. Macdonell, Journal of the Royal Asiatic Society, Volume 30, Issue 01, January 1898, pp. 117-141, DOI: 10.1017/S0035869X00146246, |url=

[1]Digital image.Kevinalfredstrom. N.p., n.d. Web. <