How Many Triangles 1

Keywords: Starter | Problem Solving | Shape

• *Judy, Chatsmore CHS
• Thursday, October 12, 2006
• "This triangle starter is excellent. I have used it with all of my ks3 and ks4 classes and they are all totally focused when counting the triangles."
• Jo Melville, Aberdeen
• Friday, June 22, 2007
• "All my S1 - S4 classes enjoyed this activity at different levels. S3 have managed to write a formula for the number of triangles in an n-row triangle. Excellent!"
• Guy Broster, Cumbria, UK
• Sunday, September 16, 2007
• "A great all-purpose starter for any lesson. Got my little ones on task and ready for their lesson."
• Mr R. Ditta, Stokesley School, North Yorkshire
• Wednesday, October 03, 2007
• "It is a great starter that could be improved if the demonstration counted along with the shading of the different triangles."
• Year 8 and Miss D, Wade Deacon High school, Widnes
• Tuesday, September 23, 2008
• "Good starter, gets the mind working. Challenging for a class of year 8's, but fun."
• Mrs Myers' Year 8 gang!, Woldgate 'the best' College
• Monday, September 29, 2008
• "My Year 8 class got 27!!! It did take us a while though! "
• Mr Sambhi, Solihull
• Tuesday, November 04, 2008
• "Excellent starter. I have used with every class I teach."
• Mr Cansdale, 7H3
• Thursday, November 20, 2008
• "We really enjoyed this activity, we want more of these please!"
• Miss Pankhurst, Norfolk
• Thursday, December 11, 2008
• "Excellent starter! I have used it with nearly all of my groups and they all got focussed on the task in hand really quickly. I also used a similar one with a 5x5 square grid and asked "how many squares?". Highly recommended!"
• Steven, University of South Carolina
• Tuesday, February 03, 2009
• "This web page is wrong there are actually 2 more than listed."
• Mrs. Happily, Norfold State Primary
• Monday, February 09, 2009
• "This was a very good activity. The students enjoyed it."
• Bill Loyd, Marysville, CA
• Sunday, February 15, 2009
• "I agree with Steve, there are 29! There are 2 coming from the 2 vertical sides that were not counted."
• Matthew, Anchorage, AK
• Sunday, February 15, 2009
• "2 more than listed?
  A great mental exercise."
• Enrique Martin, Tyler Texas
• Saturday, February 21, 2009
• "Are you sure there are two more triangles? It might be a same counted triangle looked from another perspective."
• Mike C, Grand Forks, ND
• Sunday, February 22, 2009
• "I still only count 27. Where exactly are the two missing triangles?"
• Matt, Jacksonville, Florida
• Sunday, February 22, 2009
• "Actually...there are in fact only 27. The two "vertical triangles" that you were counting...they were actually counted, but in a different perspective than it appeared."
• Rich, Florida
• Sunday, February 22, 2009
• "Steve and Bill: How do you get 29?"
• Julian, Brevard University
• Tuesday, February 24, 2009
• "There are 27. You've made the mistake of counting a "rotated" segment that when counted was viewed top to bottom. The math is true."
• Zach, Redding, Cali
• Tuesday, February 24, 2009
• "Multiply the number of lines by the number of external sides."
• Dylan, Shasta High, Redding, Ca
• Thursday, April 23, 2009
• "I happen to have written a formula for this as well. I'd post it, but I'm afraid some of the work would get quite confusing if put into text. If anyone is interested or has one to compare, Email me at YuletidePirate@hotmail.com and I'll scan mine and send it to you.
  P.S. Zach, I'm also in Redding and that's really weird that I should stumble across you. You go to Shasta?"
• Danny, Freeport, NY
• Friday, May 29, 2009
• "It's the number of internal segments or lines (do not include the sides of the whole) cubed."
• Godwin Amarikwa-Obi, St. Mary's CofE School, Hendon, London UK
• Friday, June 12, 2009
• "I used this starter as part of group qiz and my students did enjoy it. They were really thrilled when the computer counted the triangles. It is even more fun when you tell them to count 1,2,3,... at the same time as the computer. Transum.org thank you again for having wonderful 'starter of the day'."
• Tom Morley, NC
• Saturday, June 13, 2009
• "There are in fact 29. They most certainly did miss 2."
• Jared Huseman, Amarillo, TX
• Sunday, June 14, 2009
• "There are only 27 triangles. Not only do I just know this from looking at it, but come on guys. We live in the 21 century. Search Google and you'll find nothing to support this having 29 triangles."
• Jay C., Madison, Indiana
• Monday, June 15, 2009
• "There are 29. Math is not spelled with an "s"."
• Robert And Emily Bernal, California
• Thursday, June 18, 2009
• "An advertisement ask the very same question (for the triangle) but did not tell us the answer. Get this, it wanted our phone numbers. Instead we closed that page and typed in "how many triangles" into Google and found THIS page complete with the great visual answer which allowed us to count the very first time. Awesome work!"
• Hazel, London UK
• Saturday, June 20, 2009
• "Great puzzle. Maths is always spelt with an s in England - it is short for mathematics - Math is an americanism."
• Ron M, Cincinnati, Ohio
• Wednesday, July 29, 2009
• "Steve and Bill came up with 29, but if you used their same theory to count each base of a triangle, they should come up with 81 triangle bases. Which is 27 times the 3 sides for each triangle. I believe the "colour" "maths" diagam is correct with 27 triangles."
• Wil Mashburn, Oregon
• Saturday, August 01, 2009
• "I thought there were 29 also... but then I looked again and the center "3x3" triangles were already counted just rotated once."
• Megan Gibson Year 5, Marthan Primary School
• Wednesday, September 23, 2009
• "My brain wasn't switched on because I only found 20."
• Miss Jabeen, Rochdale
• Wednesday, September 23, 2009
• "My kids enjoyed this activity very much it was a great start and quite so very challenging!"
• Robbert Van Der Kruk, The Netherlands
• Sunday, October 18, 2009
• "n=4 --> 27
  n=5 --> 48
  n=6 --> 78
  In general: (n(n+1(2n+1) -n.mod(2))/8
  Proef: email robbertvdkruk@live.nl."
• S1 Lockerbie Academy, Lockerbie Scotland
• Wednesday, November 25, 2009
• "We think there are 27."
• Lionel Morbelli, My Home
• Saturday, April 17, 2010
• "2n--->n(n+1)(4n+1)/2
  2n+1->(n+1)(4n2+7n+2)/2
  which gives:
  1---->1
  2---->5
  3---->13
  4---->27
  5---->48
  6---->78
  7---->118
  8---->170
  Maybe less elegant that Robbert Van Der Kruk
  email: takelion@yahoo.com.sg
  I looked by recurrence at how many triangles are added when moving from k to k+1 "floors", where the "floors" are added below. Let S(n) be the number of triangle for n "floors":
  S(n)=S(n-1)
  +1+2+...+n <--- these are the triangles pointing up
  + (if n even) 1+2+...+n/2 +1+2+...+(n/2-1) <--- pointing down
  + (if n odd) 2(1+...+(n-1)/2) <--- pointing down
  S(n)=S(n-1) + (n even) n(3n+2)/4 + (n odd) (n-1)(n+1)/4
  where S(0)=0 and S(1)=1
  By recurrence, knowing that 1+2+...+k=k(k+1)/2 and 1+4+9+...+k2=k(k+1)(2k+1)/6, one may find the formulas of the top of this post:
  S(2n)=n(n+1)(4n+1)/2
  S(2n+1)=(n+1)(4n2+7n+2)/2."

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