Congratulations to the following students whose entries were selected as winners in the recent middle level problem solving contest. Our thanks to Mark Jensen of the mathematics department at the Melfort Comprehensive Collegiate for judging the entries. Selections were made on the basis of accuracy, creativity, detail and organization of the answers given. Problem 1 In a polygon, a diagonal is a line joining two non adjacent vertices. A triangle has no diagonals. A quadrilateral has 2 diagonals. A pentagon has 5 diagonals. How many diagonals would be in a 15 sided polygon? Winner - Lacey Walker Grade 6 , Star City School Lacey's Answer - The first thing I did to solve this problem was notice a pattern in the polygons that had an odd number of sides. A triangle has 0 diagonals, a pentagon has 5 diagonals, a 7 sided polygon has 14 diagonals, a 9 sided polygon has 27 diagonals. ( see diagrams ). This is the pattern that I discovered: 3 sided:  3 x 0 = 0 5 sided:  5 x 1 = 5 7 sided:  7 x 2 = 14 9 sided:  9 x 3 = 27 11 sided: 11 x 4 = 44 13 sided: 13 x 5 = 65 15 sided: 15 x 6 = 90 Therefore, there are 90 diagonals in a 15 sided polygon. Note from The Centralizer: There is a development of the expression behind Lacey's pattern in the Quandaries and Queries section of Math Central in a note titled Diaginals in a Polygon. Problem 2 Paul places 7 bags of marbles on a table. The second bag has 2 more marbles than the first bag. The third bag has 4 more marbles than the second bag. The fourth bag has 6 more marbles than the third bag and so on. The total number of marbles in all seven bags is 231. How many marbles are there in each bag? Winner - Rory Statham Grade 7 , Star City School Rory's Answer - First of all, to figure this question I divided 7 into 231 and got 33. Since I got 33, I knew that there couldn't be any more than 33 marbles in the first bag. So I started with a smaller number like 14 in the first bag. I used a chart and the guess and check method to find what numbers would work to get 231. My answers are 17, 19, 23, 29, 37, 47 and 59. (see chart) 1st 2nd 3rd 4th 5th 6th 7th total check 14 16 20 26 34 44 56 210 no 15 17 21 27 35 45 57 217 no 16 18 22 28 36 46 58 224 no 17 19 23 29 37 47 59 231 yes Problem 5 A pattern is made of alternating red and black disks. The first row has only one disk, the second row has three disks, the third row has five disks and so on. Each row begins and ends with a black disk. How many red disks are there in the first to fiftieth rows inclusive. Winner - D'laney Bruneau Grade 6 , Star City School D'laney's Answer - First of all, I drew a picture to figure out the question. From the picture I saw that row 1 has 0 red disks, row 2 has 1 red disk, row 3 has 2 red disks, row 4 has 3 red disks. I think this pattern will continue until row 50 has 49 red disks. Now I have to add all the numbers from 1 to 49 to get the total. I added all the numbers on my calculator to get 1225, which I think is the answer. Then I remembered learning at the beginning of grade 6 how to add a bunch of numbers in a row, like all the numbers from 1 to 49. ................... 1 + 2 + 3 + 4 + 5 + ..........+ 24 + 25 + 26 +..........45 + 46 + 47 + 48 + 49 I added the 1 and 49 to get 50. I added the 2 and 48 to get 50. I added the 3 and 47 to get 50. The adding went on like this until in the middle I added the 24 and 26 to get 50. Finally I got to 25 which didn't have a partner. Now, to get the answer I know that there are 24 pairs of 50 and the 25 by itself. So the answer is 24 x 50 = 1200. Then 1200 + 25 = 1225. Problem 6 Ryan has been collecting coins for his annual Telemiracle donation. His donation of $7.30 is made up of nickels, dimes and quarters. If he has 47 coins in total, how many of each kind of coin did he have? Is there more than one solution to the problem? Winner - Ryan Flegg Grade 7 , Star City School Ryan's Answer - To find the answer to this question I made a chart and put in the number of nickels, dimes and quarters that I thought would make $7.30. I guessed 17 nickels, 20 dimes, and 10 quarters but that was too low ($5.35), so I added 3 more nickels, left the dimes the same and took away 3 quarters but my answer got even lower ($4.75). So using common sense, I jacked up the quarters by 13 since they have greater value, left the dimes the same and took away 13 nickels. Now my answer of $7.35 was only 5 cents too high. So finally I took away 1 dime and added 1 nickel and this gave me an even $7.30. The numbers of coins that works is 8 nickels, 19 dimes and 20 quarters Nickels Dimes Quarters Total 17 20 10 $5.35 20 20  7 $4.75  7 20 20 $7.35  8 19 20 $7.30 Problem 7 Sue, the owner of Rectangular Catering Services Inc. is trying to determine the best way of seating customers in her cafeteria. The tables that are used can sit 10 people; four along each side and one at each end. She decides to join the tables together end to end. If her cafeteria has enough length for 10 tables to be placed end to end to make one row, and enough width for 5 of these rows, how many customers could be seated at one time? Winner - Rory Statham Grade 7 , Star City School Rory's Answer - I used a diagram to help me understand this problem. First I saw that when tables are put end to end they will only seat 8 people but the ones at each end would seat 9 people. If there are 10 tables in a row then there will be 10 x 8 = 80 people along the sides. Now I can multiply by 5 since there are 5 rows to get 5 x 80 = 400. Now I can add on 10 because there are 5 along the front of each row and 5 along the end of each row. So my final answer will be: 10 x 8 x 5 + 10 = 410 people. Problem 8 Beth is a walker who can walk at a pace of 5 km per hour. Lanny is a jogger who runs at a pace of 8 km per hour. Beth is an early riser and Lanny likes to sleep in. On a beautiful spring day, Beth awakens at 7 a.m. and begins her walk. Three hours later, at 10 a.m. Lanny finally awakens and begins to jog, following the same route they always use. Assuming they love to exercise at what time will Lanny catch up to Beth? Plot ordered pairs of (time, distance) to illustrate the answer. Winner - Blair Jasper Grade 9, F.W. Johnson Collegiate Regina, Saskatchewan Lanny will catch up to Beth at 15:00 hours. Return to Math Central To return to the previous page use your browser's back button.