Alphametics A cryptarithm is a type of mathematical puzzle in ..., Assignments of Reasoning

Download Alphametics A cryptarithm is a type of mathematical puzzle in ... and more Assignments Reasoning in PDF only on Docsity! 2013-­‐14  Challenge  Topic  (Meet  5,  Event  A)       Presented  at  2013  Coaches  Conference,  June  21-­‐22   Alphametics     A cryptarithm is a type of mathematical puzzle in which the digits in a mathematical equation are substituted by letters or other symbols. In a typical puzzle, the same letter or symbol always represents the same digit. The objective of the puzzle is to break the code and determine the digits used that result in a true mathematical equation. Cryptarithms may have unique solutions or several solutions. In 1955, J. A. H. Hunter introduced the word alphametic to designate a cryptarithm whose letters form meaningful words or phrases. One of the most famous cryptarithms that is also an alphametic is:   SEND                            MORE                        MONEY   This puzzle appeared in the July 1924 issue of Strand Magazine. This puzzle was made by the famous puzzlist H. E. Dudeney. The solution to this puzzle is a bit more involved than one might think. In fact, no two alphametics, even with the same operation, require the same approach to find the solution. There are, however, a few observations and standard conventions that can be applied in problem solving! Let’s start by defining the alphametic puzzle conventions. There are few basic premises: • The numbers represented by the words never contain a leading zero(s). • There is a 1-to-1 correspondence between letters and digits. • Operations are limited to +, -, *, and / . • In division puzzles, the remainder is always 0. Are there specific problem solving techniques that can be used with alphametics? There is no a set strategy for cracking the code and solving the puzzle! The answer is basic arithmetic facts, logical reasoning, solving systems of equations and determination and patience. 2013-­‐14  Challenge  Topic  (Meet  5,  Event  A)       Presented  at  2013  Coaches  Conference,  June  21-­‐22   A Few Alphametic Problem Solving Facts/Strategies 1. The operation can sometimes be deduced by the largest place value of the numbers involved or by considering intermediate steps (if given). 2. Since the digits are 0-9, the largest carry in addition with two summands is a 1. 3. In the case of K + K = K or K – K = K we know that K can only be 0 or 9 (in the case of carry 1 or borrow 1). 4. No perfect square can end with 2, 3, 7, or 8. For example, in the equation E x E = T, T cannot represent 2, 3, 7, or 8. 5. If N is even, then (N * 6 ) mod 10 = N. If N is odd, then (N * 5) mod 10 = 5. 6. The use of modular arithmetic often helps. For example, use of mod-10 arithmetic allows the columns of an addition problem to be treated as simultaneous equations, while the use of mod-2 arithmetic allows inferences based on the parity of the variables. Books: Hunter, J. A., “Challenging Mathematical Teasers,” Dover, New York, 1980 Hunter, J. A., “Mathematical Brain Teasers,” Dover, New York, 1976. Hunter, J. A., Madachy, Joseph S., “Mathematical Diversions,” Dover, New York, 1975 Web Resources: There are many web sites as well as puzzle generators and solvers on the web. Search on “cryptarithm.” Websites used to prepare materials: Interesting Variations on an alphametic (including poetry): http://www.mathematik.uni- bielefeld.de/~sillke/PUZZLES/ALPHAMETIC/alphametic-mike-keith.html#trad A cryptarithm creator and solver: http://www.iread.it/cryptarithms.php