Elementary POW, February 26 - March 1, 1996
Elementary POW Problems || January-March, 1996 Problems || Elementary POW Main Page
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Elementary Problem of the Week, February 26 - March 1, 1996
This week's problem and Bonus Puzzler both involve hexagons and the number
six.
There are six people at a party. Each person shakes hands with each of the
other guests. How many handshakes take place?
(Hint: Draw a hexagon. Let each corner of the hexagon represent one of the
guests. How can this diagram help you solve this problem? What geometric
shape would you draw if there were eight people at the party? Would this
method still work?)
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Bonus Puzzler
Seven pennies can be laid out to form a hexagon in which each side is two
pennies long.
o o
o o o
o o
For this reason, the number 7 is called a "hex number." Can you find the
next two hex numbers? (Each side on the next one will be three pennies
long, and so on.)
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Great job on this week's solutions. We have many new teams joining us along with new students from schools that have been active participants. A special welcome to Amy Forster, a home-schooled student from Cygnet, Tasmania, Australia and Anna Margush, a 4th grade home-schooled student from Akron, Ohio. Also, in reply to Mike Bernstein, a 4th grader at Georgetown Day School in Washington D.C, who wrote, "Teacher: Paul Nass, I think he's famous by now. Is his name on here alot?" Mr. Paul Nass is certainly famous from my perspective. We can hardly keep up with all the solutions sent to us by his students. Keep up the great work, Mr. Nass. Your students are lucky to have such a fine, dedicated teacher. Be sure to read the Blast-Out Math Maniacs Club, Olivehill Accelerated School, Dayton, OH, solution. They included a list of who they would invite to shake hands with. A note of congratulations to Mr. Crandall's Science Lab class, Grade 6, Deefield, Ks. Middle School who wanted to challenge themselves and correctly solved the bonus puzzler for a hex number with 10 sides! -Ruth ******************* Amy Forster, I'm in grade 7. (POW * Bonus) I do home school and teach myself. I live in Cygnet,Tasmania,Australia,7112 ******************* Anna Margush (POW & Bonus) Grade 4 Home School Akron, Ohio ************* Ryan Grace, Kim Fugok, Christine McGowan, and Amanda Tumminelli (POW & Bonus) Mrs. Brennan - Grade 6 Drexel Hill School of the Holy Child, Drexel Hill, PA ****************** Mr. Crandall's Science Lab class, (POW & Bonus) Grade 6 Deefield, Ks. Middle School ************* Julia C., Nick S., Mike D., Meghan R., Meghan L. Grade 4 (POW & Bonus) Miss Seager Bagnall School, Groveland, MA ***************** Matt Spindler, Evan Mossman, and Todd Feiler (POW & Bonus) Mr. Fred Rimmel - Grade 4 Kerr Elementary Pittsburgh, PA ********************** Shauntea Davis, 4th grade (POW) Blast-Out Math Maniacs Club Olivehill Accelerated School Dayton, OH 45426 ******************** Jamie and Kelly Detzel (POW) Saxonburg Elementary School Saxonburg, PA. Kelly is in grade 4 and Jamie in 6th grade. **************** Ted Powers (POW), Chris Wade and Liam Aylward (POW) & (Good try on Bonus), Lauren Clark (POW), Mrs. Pensa's 3rd Grade class Center School Stow, MA ****************** Natalie Moore (POW) Miss Mc Carthy's 4th Grade Class Center School Stow, MA ********************** Rob, Jonathan B., Brandon H., Randy, Brian (POW) Grade 4 - Priscilla Roehm Mandarin Oaks Elementary Jacksonville, FL ******************* Name: Mike Bernstein (POW), Kate (POW), Becky Cisin (POW), Grade: 4 Teacher: Paul Nass, I think he's famous by now. Is his name on here alot. School: Georgetown Day School Place: Washington D.C **************** Jacyln Cohen (POW), Aneil (POW) Grade 5 Teacher: Paul Nass, School: Georgetown Day School Place: Washington D.C ********************* Mrs. Bach's 5th Grade Class (POW) Joyce Kilmer School Mahwah, NJ *************** Munsey Park School Manhasset, NY Mrs. Hirns class in fourth grade. Matt and Jeffery (POW) ***************** Pope John XXIII Catholic School ***************** Christopher, Mike, and Shane (POW) 3rd Grade Mrs. Kaye's class Lincoln Elementary School Burlingame, CA **************** Anna Colom (POW) Grade 4 Mrs. Hamilton Bagnall School, Groveland, MA ****************** Meagan (POW), Ashley (POW), Lindsay (POW), Alan (POW), Rick & Billy (POW & Good try on Bonus) Grade 6 Mrs. Caruso Bagnall School, Groveland, MA ******************** Mark Mrs. Crawford Grade 5 Bagnall School, Groveland, MA ********************* Matt, Dan and Patrick (POW) Miss Duggan's fourth grade class Munsey Park School Manhasset, NY ************* Jessica Verrilli Mrs. McNabb. - 5th grade St. Joseph School Seattle, Washington ****************** Jonathan L. and Alp 4th Grade Mandarin Oaks Elementary Jacksonville, FL *************** Dr. Slomer's fifth grade class Rogers elementary School Pittsburgh, Pa. *************** Kirk Thompson (POW & Bonus), John Tiernan (POW), Jenna Gilbert (POW), Allison Cullen Doyle (POW), Stephanie Catanese (POW), Brad Jadlowiec (POW), Christine Burgander (POW) *************** High School Math Mentors at Shaler High School: Julie Jadlowiec Ellen Charney Keith Monteleone Mark Bernebrg Mike Pschirer Nick Szmyd ***************
********************* Hi! My name is Amy Forster,and I'm in grade 7.I do home school and teach myself.I live in Cygnet,Tasmania,Australia,7112.I love maths and really enjoyed doing your puzzles! From, Amy Forster Answer to this week's problem:15 hand shakes between 6 people Solution: For 6 people at the party: A BPerson A shakes hands with B,C,D,E,F=5 hand shakes. FC Person B shakes hands with C,D,E,F=4 hand shakes. E D Person C shakes hands with D,E,F=3 hand shakes. Person D shakes hands with E,F=2 hand shakes. Person E shakes hands with F=1 hand shake. 5+4+3+2+1=15 hand shakes between 6 people. For 8 people at the party: A BPerson A shakes hands with B,C,D,E,F,G,H=7 hand shakes. HCPerson B shakes hands with C,D,E,F,G,H=6 hand shakes. GDPerson C shakes hands with D,E,F,G,H=5 hand shakes. F EPerson D shakes hands with E,F,G,H=4 hand shakes. Person E shakes hands with F,G,H=3 hand shakes. Person F shakes hands with G,H=2 hand shakes. Person G shakes hands with H=1 hand shake. 7+6+5+4+3+2+1=28 hand shakes between 8 people. If there were n people at the party,the number of hand shakes would= (n-1)+(n-2)+(n-3)+...................+3+2+1. I am very excited to have found these problems on the internet.I can't wait until next week's!! Answer to Bonus Puzzler:The next 2 hex numbers are 19&37. Solution: It took a lot of trial and error for me to work out the relation ship between the number of rows and the hex number,but I did it in the end(I think)! Row numberRelation shipHex number 11((3x1)-2)-(1-1)=11 22((3x2)-2)-(2-1)=77 33((3x3)-2)-(3-1)=1919 44((3x4)-2)-(4-1)=3737 Basic relation ship= row no((3xrow no)-2)-(row no-1)=Hex no. ie if n=number of rows H=Hex number H=n(3n-2)-(n-1) ********************* ******************** >Ryan Grace, Kim Fugok, Christine McGowan, and Amanda Tumminelli - Grade 6 >Drexel Hill School of the Holy Child, Drexel Hill, PA Mrs. Brennan (POW & Bonus) > >We acted out a simpler problem with four students since we have four students >in our group then we drew a chart which is drawn below. > > >1 2 3 4 5 6 > >1 - x x x x x > >2 - - x x x x > >3 - - - x x x > >4 - - - - x x > >5 - - - - - x > >6 - - - - - - > > >The X represents each handshake. So, if six people at a party shake hands with >each other then there would be 15 handshakes. > > >BONUS > >We used circles instead of pennies to find out that the next two hex numbers >would be 19 and 37. We noticed that the center row increased by two pennies >each time. So, for 19 we added 5 + 4 + 4 + 3 + 3 = 19 and for 37 we added 7 + >6 + 6 + 5 + 5 + 4 + 4 = 37 > *************************** Elementary Problem of the Week, February 26 - March 1, 1996 Anna Margush Grade 4 Home School Akron, Ohio margush@uakron.edu Problem I solved this two ways. There are 6 people at the party. Way 1: Person 1 shakes hands with 5 people Person 2 shakes hands with 4 other people (he already shook with person 1) Person 3 shakes hands with 3 other people (he already shook with persons 1 and 2) and so on until Person 6 is reached. He does not need to shake any more hands. There are 5+4+3+2+1 handshakes or 15 in all. Way 2: Since each person shakes hands with 5 others, so there are 5*6 or 30 handshakes. But when person 1 shakes hands with person 2, it is the same as person 2 shaking with person 1. So, if we divide 30 by 2, we get 15 handshakes. If there were 8 people at the party you would use an octagon Bonus Puzzler By arranging pennies on the floor, I was able to find that the next two hex numbers are 19 and 37. ************************************** Mr. Crandall's Science Lab class, Deefield, Ks. Middle School-Grade 6 Answer to Problem - 15 Person 1 shook hands with persons 2,3,4,5, & 6. Person 2 shook hands with persons 3,4,5,& 6. Person 3 shook hands with persons 4,5,& 6. Person 4 shook hands with persons 5 & 6. Person 5 shook hands with person 6. Total of 15 hand shakes. Bonus puzzler answers. The next two "hex numbers" are 19 and 37. For 19 you add 3+4+5+4+3. For 37 you add 4+5+6+7+6+5+4. We really challenged ourselves and found out the hex number if 10 pennies were on each side. We came up with 271. Is that Correect? *********************** My name is Jaclyn Cohen. I am in the 5th grade. My teacher's name is Paul Nass. I go to Georgetown Day School in Washington, DC. My answer is: 1) say that each kid was a number 2) there is kid#1, kid#2, kid#3, kid#4, kid#5, and kid#6 3) kid#1 would shake hands with #2, #3, #4, #5, and #6 4) that is 5 hand shakes 5) kid #2 would shake hands with #3, #4, #5, and #6 6) that is 4 hand shakes 7) kid #3 would shake hands with #4, #5, and #6 8) that is 3 hand shakes 9) kid #4 would shake hands with #5 and #6 10) that is 2 hand shakes 11) kid #5 would shake hands with #6 12) that is 1 hand shake 13) add up all the hand shakes and it equals 15 hand shakes!!! 14) you can't shake hands with yourself and you can't shake hands with a person twice, example: #3 shakes hands with #4, #4 can't shake hands with #3!!!! ********************** Hi our names are Jamie and Kelly Detzel. We go to Saxonburg Elementary School in Saxonburg, PA. Kelly is in grade 4 and I am in 6th grade. We got the problem of the week . There are 2 possible answers. We will explain each and then tell you which one we think is better. One answer is 30. If you asked each person how many hands they shook they would say 5. Since 6 times 5 = 30 we have one answer. HOWEVER, this number includes each person shaking anothers hand and both people counting it as a handshake. This might be true but then each shake is counted twice. A better answer would be 15. Here is a way to figure out how many handshakes with any number of people. Say d is the number of people. It could be any number at all. Multiply d times one less than d then divide the answer by 2. This gives the number of handshakes. Jamie and Kelly **************** Julia C., Nick S., Mike D., Meghan R., Meghan L. Grade 4 Miss Seager Bagnall School, Groveland, MA We drew a hexagon. Each corner represented a person. We had number 6 shake hands with everyone else and drew lines to the other corners to show whose hands were shaking. Number 6 made 5 hand shakes. The second guest, number 5 made 4 hand shakes. The third guest shook 3 hands. The forth guest shook 2 hands and the fifth guest shook 1 hand. That makes 15 hand shakes. Bonus Puzzler: Meghan R. found 19 pennies in the second hex number and 37 pennies in the third hex number. *********************** Matt Spindler, Evan Mossman, and Todd Feiler Grade 4 Mr. Fred Rimmel Kerr Elementary Pittsburgh, PA We arrived at the answer of 15. We got the answer by drawing dots on the marker board and connecting them with lines that stood for hand shakes. Then we found out that the lines formed a hexagon. Another way to solve the problem is you can start out with 5 and subtract 1 each time. Then you add them together. If you did it with 8 you would get an octagon. Bonus: The next two hex numbers are 19 ( sides of three) and 37 ( sides of four). Fred Rimmel Kerr Elementary Pittsburgh ******************** Our math club acted out the handshakes problem. We used a Bugs Bunny stuffed pillow as a person because only five people showed up tonight. Here's what we found out. 1st person = 5 handshakes 2nd person = 4 handshakes 3rd person = 3 handshakes 4th person = 2 handshakes 5th person = 1 handshake 6th person = 0 handshakes All the handshakes added up to 15 handshakes for 6 people. We even tried some other problems for fun. We found 10 people would have 45 handshakes and 3 people would have 3 handshakes. Then we started to use our imagination. We thought the most awesome way to figure this out would be with guest stars. We would invite the following people: Tiffany-Amber Thiessen (actress on 90210-Valerie) Johnathon Taylor Thomas (actor on Home Improvement) Mary J. Blige (singer) Chili (singer with TLC) Tommy (the Power Ranger) Romeo (singer in Immature) This would be the best problem to solve if they were here to help! Do you think we have a chance of them coming? Shauntea Davis, 4th grade Blast-Out Math Maniacs Club Olivehill Accelerated School Dayton, OH 45426 ****************************** Lauren Clark in Mrs. Pensa's class at Center School Stow MA > >In order to find the answer I first drew a hexagon and numbered the corners 0 >- 5. I first had 5 shake hands with 4, 3, 2, 1, and 0. Then I had 4 shake >hands with 3,2,1, 0 but not 5 since they already shook hands. I then had 3 >shake hands with 2,1, and 0 but not 4 and 5...and so on. So my formula for >this problem is: > >5 + 4 + 3 + 2 + 1 = 15 because # 5 shook hands with 5 people and 4 shook hands >with 4 people etc.. ************************* Rob, Jonathan B., Brandon H., Randy, Brian Grade 4 Priscilla Roehm Mandarin Oaks Elementary Jacksonville, FL 15 handshakes First we gathered people to be in the group. Then we all got in a circle. Next we shook each other's hand. Finally, we counted how many handshakes and it was 15. And that's how we got the answer. ********************** >Natalie moore miss mccarthy Center school Stow Mass. > >Problem 1-First I drew a hexagon and then I drew dots too represent the >people. So I counted the hand shakes. Each one was 5 then 4 then 3 then 2 then >1. So then I counted them up. I got 16. The diagram helped me by planning it >out and not do it in my head and I know I wont get it wrong. if there were >eight people at the party I would draw a octagon. Yes the method would still >work. ******************************* It's me, Jaskimack. The answer to the problem of the week is 15 handshakes. The way I got this was: I drew six circles and formed them into a hexagon shape. I picked 1, and drew a line from that circle to all of the other circles. I did the same with the next, but I didn't draw a line to the first one that I had drawn lines with. I repeated this, each time taking away one more of the circles that I had drawn lines with. Name: Kate Grade: Fourth Teacher: Paul School: Georgetown Day School Location: 4530 MacArthur Blvd., NW Washington, DC 20007 ********************* Ted Powers - Mrs. Pensa's 3rd Grade class - Center School Stow MA Answer 15. I found out my answer because the first person shook 5 people's hands. The second person shook 4 people's hands all the way down to 0. So 5+4+3+2+1+0=15. *********************** >Chris Wade and Liam Aylward in Mrs. Pensa's 3rd grade class Center School Stow >MA > >First we made the first corner shake hands with everybody, that was 5 >handshakes. Then we made the second corer shake hands with everyone except the >first person, that was 4 handshakes. And all the way down to when someone >could only shake one persons hand because he already shook hands with everyone >else. Add 5+4+3+2+1=15, our answer. ************************* Mrs. Bach's fifth grade class, Joyce Kilmer School Mahwah, NJ found this handshake problem very easy--there were 15 handshakes. It was easy for us because in September we had to figure out how many handshakes there would be if 100 people were in the room each shaking hands. Maybe the other kids who do these POW's would like to try that. Imagine you are at a party. There are 100 people in all. If everyone at the party shakes hands once with everyone else, how many handshakes will there be in all? **************** The answer is 15 handshakes. Because... A shakes B,C,D,E,and F=5 handshakes B shakes C,D,E,and F=4 handshakes C shakes D,E,and F=3 handshakes D shakes E,and F=2 handshakes E shakes F=1 handshakes F doesn't shake anybody's hand. 5+4+3+2+1+0=15 so there were 15 handshakes made at the party. Christopher, Mike, and Shane 3rd Grade Mrs. Kaye's class Lincoln Elementary School ************************ My name is Jessica Verrilli. I'm in fifth grade at St. Joseph School. St. Joseph School is located in Seattle, Washington. My teachers name is Mrs. McNabb. The math problem that I figured out was: There are 6 people at a party, each shakes hands with each other. How many hand shakes are there? I figured out that there were 15 handshakes. The way I figured it out was: There are 6 people. Each shakes hands with 5 others. 6 x 5 is 30. But each hand shake consists of two people. Therefore, you divide 30 by 2. You get 15 hand shakes. ******************* Jonathan L. and Alp 4th Grade Mandarin Oaks Elementary Jacksonville, FL 15 handshakes Step 1 Draw a hexagon with six dots Step 2 Take one dot and line up to the other dots Step 3 Cross out that dot Step 4 Do step 2 except except don't line up the dot you crossed out Step 5 Do step 3, then step 2 and so on until you run out of dots **********************
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