Rearrange the matches - October 16-20, 1995
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These thirteen matches are arranged to make six equal regions. Take away one of the matches and arrange the remaining twelve so that they still make six equal regions.
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Annie says: Solutions
Good job this week!
One thing that came up in some of the solutions is what restrictions there are that aren't stated. For example, since we're using matches, to me that means that all the segments involved in our solution must be the same length. Now, if I had said 'sticks', then if I wanted them all the same length, I'd have to say so. But I think that when one uses matches or toothpicks, equal lengths are implied.
While most people went with the hexagon answer, there were a couple of alternative two-dimensional solutions, and then there were a few people who came up with the cube, which does indeed have twelve edges and six faces. Excellent! That hadn't occurred to me right off - often when you get one answer, you don't bother to think about others, especially if you know you're right. But there isn't always just one right answer.
Clayton Grondzik gave an alternative interpretation of 'region', and showed that he has a sense of humor (sometimes very important in mathematics). Carolyn DiMaria and Kathleen Wuerth pointed out that perhaps matches aren't the safest thing to be playing with, and suggested toothpicks. :-)
Following are highlights; this week we welcome Sweden and Northern Ireland. The names of the people who submitted correct solutions and their solutions are also available.
Melaina Brown Grade 9 School: Sammamish High School M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M I got this configuration of matches by thinking inductively and rearranging the matches many times until I got the right shape with the right number of matches and regions.
Clayton Grondzik Grade: 8 School: Center for the Arts and Sciences My best answer: ********* * * * * * * * * * * * * ***************** * * * * * * * * * * * * ********* Twelve matches, all regions are equal. My favorite answer? This on depends on how loosely you define the word region. ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* ********* All regions are equal (=). Get it?
Dylan Matheson School: Martin County High School, Stuart, Florida :--- |\ |\ | \ | \ |___|___. \ |\ | \ | \ | \|__\| This contains 12 matches and forms 6 areas of equal size.
Ryan Carpenter Grade: 8 School: Murray Junior High School There are many way to do this problem and here is one way I think works (each horizontal line equals one matchstick - one matchstick on the top, two matchsticks on the bottom). This object can be rotated and rearranged. MMMMMMMMMMMMMMMMMMMM M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M MMMMMMMMMMMMMMMMMMMM MMMMMMMMMMMMMMMMMMMM >Neat, Ryan. I didn't think of this one! Nice job. >Can you see another way to do it? > -Annie Annie, I just found an answer this morning as I was reading your letter, and here it is: MMMMMMMMMMMM M M M M M M M M M M M M MMMMMMMM MMMMMMM M M M M M M M M M M MMMMMMMMMM The matchsticks can be the same length.
Stefan Axelsson Year 7 School: Grundskolan, Gothenburg, Sweden If everything goes right there should be a gif with a picture of the solution added to this letter. Nock on wood! [and there was, and it was excellent! - Annie]
Matthew Davis Age 12 School: Friends' School, Lisburn, Northern Ireland The diagram is a regular hexagon and has a match going into the centre from each corner, i.e., with the diagonals drawn in.
Carolyn DiMaria and Kathleen Wuerth Grade 9 School: Mount St. Joseph Academy, Flourtown, PA The problem of the week for this week wasn't really challenging. They gave you thirteen matches arranged to form six equally shaped figures. The example was a rectangle with three matches on the top and three on the bottom, with seven in between the top and bottom. The ones in between were placed at both ends of the top and bottom matches and in the exact middle of them. Make sense? It looks kind of like this ___ ___ ___ I I I I I I I Sorry, I'm not real good with computers but ___ ___ ___ you get my drift. If you only have twelve matches, it is still possible to make six equally shaped figures. There are three different ways that we found to solve this. One way we figured out is making a hexagon, which uses six matches for the perimeter. Then, put a match in each of the six interior angles of the hexagon so that they all meet in the middle at one point. That uses the remaining six matches and you have six equal figures. We just drew little sticks on paper and after several tries found the answer. I usually do what the problem says, like I painted figures and rolled them across paper and actually made a cube and painted its faces. BUT I WOULD NOT USE MATCHES TO SOLVE THIS PROBLEM! Use toothpicks to experiment and be a little safer.
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