Dictionary Day (Oct. 16) Meeting (Multiple Topics) Topic This meeting s topics cover a wide range of math topics, though every problem is word- or letterrelated. The majority of the problems use general math or logic, with some counting, symmetry and number theory included. Materials Needed Copies of the Dictionary Day problem set (Problems and answers can be viewed here, but a more student-friendly version in larger font is available for download from www.mathcounts.org on the MCP Members Only page of the Club Program section.) Calculators Meeting Plan This meeting is perfect for Dictionary Day (Oct. 16) because, though it involves a lot of math, every problem is related to words or letters. The first eight problems in the problem set are relatively straightforward; students should be able to successfully attempt/tackle them if working in pairs or individually. Problem #9 could require more explanation from the coach, and problem #10 is designed to be a nice wrap-up activity for the entire club to do together. 1. The letters of the alphabet are each assigned a random integer value, and H = 10. The value of a word comes from the sum of its letters values. If MATH is 35 points, TEAM is 42 points and MEET is 38 points, what is the value of A? 2006 School Competition, Team Round #6 2. The word-sum of a word is determined by adding together the value of each letter. In the alphabet, letters A through H each have a value of 5 cents; letters I through R each have a value of 7 cents; and letters S through Z each have a value of 8 cents. What is the word-sum of the word MATHCOUNTS? 2005 School Competition, Sprint Round #1 3. How many letters of the alphabet shown below have a vertical line of symmetry? 2005 School Competition, Sprint Round #7 ABCDEFGHIJKLMNOPQRSTUVWXYZ 4. A set of magnetic letters contains two of each consonant and three of each vowel. Only a complete set of letters may be purchased. How many complete sets of magnetic letters must be purchased to make a sign that reads: MATHCOUNTS COMPETITION TODAY? 2002 School Competition, Team Round #1 5. The sequence below is formed by repeating the letters of the alphabet, in order, the same number of times as the letter s ordinal position in the alphabet. After 26 Zs, the sequence starts over with ABBCCC. What is the 2005th letter in the sequence? A B B C C C D D D D E E E E E 2005 School Competition, Team Round #8 6. If all of the letters of the word BEEP are used, in how many different ways can the four letters be arranged in a four-letter sequence? The two Es are indistinguishable, so EEPB should be counted only once since we would not be able to tell a difference if the two Es were swapped. 2007 2008 School Handbook, Volume I, Warm-Up 4-3 (For #7, #8) A value is assigned to each letter of the alphabet such that A = 1, B = 2, C = 3,..., Z = 26. The wordproduct of a word is the product of the values of each letter in the word. For example, the word-product of NAME is 14 1 13 5 = 910. 7. What is Carrie s favorite breakfast food if its word-product is 4655? 8. What is her favorite dessert if its word-product is 165? 2008 2009 MATHCOUNTS Club Resource Guide 25 Club Resource Guide.pdf 25 8/18/08 11:24:15 AM
9. A value is assigned to each letter of the alphabet such that A = 1, B = 2, C = 3,..., Z = 26. A nine-digit code is then created for each letter using the prime factorization of its assigned value. The first digit of a letter s code is the number of times 2 is used as a factor, the second digit is the number of times 3 is used as a factor, the third digit is the number of times 5 is used as a factor and so on. For example, since N is the 14 th letter of the alphabet and N = 14 = 2 1 7 1, the code for the letter N is 100100000 with 1s in the first and fourth positions because its prime factorization has one 2 (the first prime number) and one 7 (the fourth prime number). What 6-letter word does the following sequence of six codes represent? The first row is the code for the first letter of the word, the second row is the code for the second letter of the word and so on. 2006 Chapter Competition, Target Round #6 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 10. There are many words that can be made from the 10 letters in the word DICTIONARY. According to the definition of word-product given for #7 and #8, what word can you make with the highest word-product? Answers: 21 points; 68 cents; 11 letters; 3 sets; V; 12 ways; EGGS; CAKE; EQUALS; multiple answers the word TRY has a word-product of 20 18 25 = 9000, so incorporating these three letters will give multiples of 9000. Possible Next Steps Using the definition of word-product, students can work on #10 individually for a little while and then as a group try to figure out the highest-scoring word they can think of. **Please send us the highest-scoring word your club finds, and we will share some of the top-scoring words online. Be sure to give us your school name when you submit your high-scoring word and send it to info@mathcounts.org with subject line MATHCOUNTS Club Program. Similarly, students can come up with riddles like #7 and #8. Discuss why some words would be much more difficult to guess (many non-prime factors) and others would be much easier to figure out (only prime factors). Perhaps give students some words and ask them to provide their respective codes using the instructions for problem #9. 26 2008 2009 MATHCOUNTS Club Resource Guide Club Resource Guide.pdf 26 8/18/08 11:24:15 AM
A = 1 Dictionary Day Meeting Problem Set A = 1 1. The letters of the alphabet are each assigned a random integer value, and H = 10. The value of a word comes from the sum of its letters values. If MATH is 35 points, TEAM is 42 points and MEET is 38 points, what is the point value of A? 2006 School Competition, Team Round #6 2. The word-sum of a word is determined by adding together the value of each letter. In the alphabet, letters A through H each have a value of 5 cents; letters I through R each have a value of 7 cents; and letters S through Z each have a value of 8 cents. What is the wordsum of the word MATHCOUNTS, in cents? 2005 School Competition, Sprint Round #1 3. How many letters of the alphabet shown below have a vertical line of symmetry? 2005 School Competition, Sprint Round #7 ABCDEFGHIJKLMNOPQRSTUVWXYZ 4. A set of magnetic letters contains two of each consonant and three of each vowel. Only a complete set of letters may be purchased. How many complete sets of magnetic letters must be purchased to make a sign that reads: MATHCOUNTS COMPETITION TODAY? 2002 School Competition, Team Round #1 5. The sequence below is formed by repeating the letters of the alphabet, in order, the same number of times as the letter s ordinal position in the alphabet. After 26 Zs, the sequence starts over with ABBCCC. What is the 2005th letter in the sequence? A B B C C C D D D D E E E E E 2005 School Competition, Team Round #8 6. If all of the letters of the word BEEP are used, in how many different ways can the four letters be arranged in a four-letter sequence? The two Es are indistinguishable, so EEPB should be counted only once since we would not be able to tell a difference if the two Es were swapped. 2007 2008 School Handbook, Volume I, Warm-Up 4-3 (For #7, #8) A value is assigned to each letter of the alphabet such that A = 1, B = 2, C = 3,..., Z = 26. The word-product of a word is the product of the values of each letter in the word. For example, the word-product of NAME is 14 1 13 5 = 910. 7. What is Carrie s favorite breakfast food if its word-product is 4655? Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Problem Set
8. What is her favorite dessert if its word-product is 165? 9. A value is assigned to each letter of the alphabet such 0 0 1 0 0 0 0 0 0 that A = 1, B = 2, C = 3,..., Z = 26. A nine-digit code is then created for 0 0 0 0 0 0 1 0 0 each letter using the prime factorization of its assigned value. The first 0 1 0 1 0 0 0 0 0 digit of a letter s code is the number of times 2 is used as a factor, the 0 0 0 0 0 0 0 0 0 second digit is the number of times 3 is used as a factor, the third digit 2 1 0 0 0 0 0 0 0 is the number of times 5 is used as a factor and so on. For example, since N is the 14 th letter of the alphabet and N = 14 = 2 1 7 1 0 0 0 0 0 0 0 1 0, the code for the letter N is 100100000 with 1s in the first and fourth positions because its prime factorization has one 2 (the first prime number) and one 7 (the fourth prime number). What 6-letter word does the following sequence of six codes represent? The first row is the code for the first letter of the word, the second row is the code for the second letter of the word and so on. 2006 Chapter Competition, Target Round #6 10. There are many words that can be made from the 10 letters in the word DICTIONARY. According to the definition of word-product given for #7 and #8, what word can you make with the highest word-product? **Answers to these problems are on page 26 of the 2008-2009 Club Resource Guide.** Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Problem Set
Dictionary Day Solutions (2008-2009 MCP Club Resource Guide) Problem 1. Since H = 10, from MATH = 35 points, we know MAT = 35 10 = 25 points. From TEAM being 42 points, we can see that the E = 42 25 = 17 points. Notice that MEET and TEAM only have one letter different. Since MEET is 38 points and TEAM is 42 points, the difference in their points is 42 38 = 4 points. This means an A is 4 points more than an E. Since we know E = 17 points, then A = 17 + 4 = 21 points. Problem 2. According to the values given, MATHCOUNTS will have a word-sum of 7 + 5 + 8 + 5 + 5 + 7 + 8 + 7 + 8 + 8 = 68 cents. Problem 3. In the alphabet shown, the letters A, H, I, M, O, T, U, V, W, X and Y have vertical lines of symmetry. This is 11 letters. Problem 4. In the sign MATHCOUNTS COMPETITION TODAY, there are multiple copies of many letters. Here is our total tally: M 2; A 2; T 5; H 1; C 2; O 4; U 1; N 2; S 1; P 1; E 1; I 2; D 1; Y 1. The consonant with the highest total is T. We would need 3 complete sets to get our 5 Ts. The vowel with the highest total is O, but we would only need 2 complete sets to get our 4 Os. Therefore, we will need 3 complete sets of letters to make the sign. Problem 5. Let s determine how many letters are written when we do one complete cycle of the alphabet. Notice there will be 1 A and 26 Zs, there will be 2 Bs and 25 Ys, etc. Each pairing adds to 27 letters, and there will be 13 pairings. This is a total of 351 letters. Dividing 2005 by 351, we see that we will have written 5.7 cycles of the alphabet. Six cycles of the alphabet will put us at 351 6 = 2106 letters. Subtracting off 26 Zs we have 2106 26 = 2080 letters with the last being Y. Subtracting off 25 Ys we have 2080 25 = 2055 letters with the last being X. Subtracting off 24 Xs we have 2055 24 = 2031 letters with the last being W. Subtracting off 23 Ws we have 2031 23 = 2008 Vs. Since there are 22 Vs, we can see that the 2005th letter will be a V. Problem 6. Since our Es are indistinguishable, we can easily list out the possibilities for the placement of our Es in a four-letter sequence: EE E E E E E E E E EE With each of these six sequences, we can place the B and P in two ways. For instance EE can become either EEBP or EEPB. Therefore, there are a total of 6 2 = 12 ways to arrange the four letters if the Es are indistinguishable. Problem 7. The word-product 4655 can be factored down to 5 7 7 19. The letters corresponding with 5, 7 and 19 are E, G and S, respectively. Notice that multiplying any of these factors together will give us more factors of 4655, but they will be greater than 26 and not correspond to a letter. Since 1 is a factor of every number, it s possible we can use an A. However, at this point we have E, G, G and S, and without needing to rearrange these, we gets EGGS, which is a well-known breakfast food. Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Solution Set
Problem 8. The word-product 165 can be factored down to 5 3 11. The letters that correspond with these values are E, C and K. Again, remember that A can always be added. It s possible to break 165 into 11 15, but this only allows us to work with K and O, and possibly A. Using E, C, K and A we can get the popular dessert of CAKE. Problem 9. The values 1 through 26 are assigned to A through Z, respectively. A 9-digit code is created for each letter using prime factorization. The first digit of a letter s code is the number of times 2 is used as a factor; the second digit is the number of times 3 is used as a factor and so on. We are given six 9-digit codes and asked to determine what word this set of codes spells. The first 9-digit code is: 001000000. The only 1 is in the 3rd column where the 3rd prime, 5, is used as the factor, so 5 is the first value or E. The second 9-digit code is: 000000100. The only 1 is in the 7th column where the 7th prime, 17, is used as the factor, so 17 is the second value or Q. The third 9-digit code is: 010100000. The 1s are the second and fourth prime, or 3 and 7. 3 7 = 21 or U. The fourth 9-digit code is: 000000000. The only number with no prime factors is 1 or A. The fifth 9-digit code is: 210000000. 2 2 3 = 4 3 = 12 or L. The sixth 9-digit code is: 000000010. The only 1 is in the 8th column where the 8th prime, 19, is used as the factor so 19 is the value or S. Our word is EQUALS. Problem 10. Answers will vary. Copyright MATHCOUNTS, Inc. 2008. MATHCOUNTS Club Resource Guide Solution Set