Can You Think of a False Statement that Appears True?

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Discussion Overview

The discussion revolves around identifying statements that may seem true or intuitive but are actually false. Participants explore various examples and problems, including mathematical puzzles and sequences, while also engaging in light-hearted commentary about teaching methods and perceptions of truth in mathematics.

Discussion Character

  • Exploratory
  • Homework-related
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant asks for examples of statements that sound true but are false, not limited to geometry.
  • Another participant humorously suggests that certain introductory phrases can increase the likelihood of engaging responses in discussions.
  • A participant presents the "missing area problem," detailing how the number of areas created by secants connecting points on a circle can be calculated, emphasizing the non-uniform spacing of points.
  • There is a playful exchange about sequences, with one participant referencing a sequence of numbers and humorously suggesting an unexpected next number.
  • A participant shares a personal anecdote about a teacher's trick question involving subway stop numbers, illustrating how context can alter expected answers.
  • Concerns are raised about a specific example involving triangles, with one participant questioning the similarity of two triangles based on their dimensions and gradients.
  • Another participant clarifies that the triangles in question are not similar, suggesting that their appearance may be misleading.

Areas of Agreement / Disagreement

Participants express differing views on the nature of the problems presented, with some engaging in playful banter while others focus on the mathematical accuracy of examples. There is no consensus on the validity of the triangle example, as some participants challenge the assumptions made.

Contextual Notes

Some statements rely on specific interpretations of mathematical concepts, and the discussion includes unresolved questions about the properties of the triangles mentioned. The exploration of the "missing area problem" also leaves open the exact formula and its derivation.

Saad
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Can you think of a statement, not necessarily in geometry, which sounds true, appears to be true in some cases, but is actually false.
 
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Homework by any chance? Generally prefacing your question by the phrases

1. this is interesting...
2. hey, can anyone show...
3. now I fouund this neat thing, can some one explain it...

increase the chances "exponentially"
 
It is intuitively obvious that . . .
 
Here's a cute, well known problem (it's referred to as the "missing area problem").

Take a circle, and mark n points around the circumference of the circle NOT evenly spaced. Draw all secants connecting the points. How many areas does that divide the circle into? (If n were even and you spaced all points equally, all "diameters" would cross at the center. Moving some of the points slightly will move some secants off the diameter so you will have line crossing near the center, increasing the number of areas. The point of "NOT evenly spaced" is to get the maximum number of areas.)

For example if there is only one point, there are no secants and the circle remains 1 area.
If there are two points, there is one secant and the circle is divided into two area.
If there are three points, there are 3 secants, forming a triangle. There is the area inside the triangle and 3 areas between the there sides and the circle: total of 4 areas.
If there are 4 points, there are 6 secants, four forming a quadrilateral and 2 diagonals. The two diagonals divide the quadrilateral into 4 areas and we have the four areas between the sides and the circle: total of 8 areas.
If there are 5 points, it's harder to count but there are 16 areas.

There is, believe it or not a simple formula for the number of area given n points!

How many areas if there are 6 points? You can draw a picture and count the areas or use the formula: there are 31!

(Hey, I said it was a "missing area" problem.)

In case you were wondering with 7 points, you get 57 areas.
 
Ok I like that one HallsofIvy. So next time someone gives me an annoying "1, 2, 4, 8, 16 ... what comes next question, then I can say 31. Nice one. :)
 
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I once had a teacher who gave us the sequence "35, 34, 33, 32, 31, 30, 29,..." and asked "what comes next". The correct answer? "61", of course!

(These were the numbers on the subway stops on his way into work. Between stop "29" and "28", the train switch on to a different line.)
 
I just did a search on google on 'missing area' and came up with this page:

http://www.reed.edu/~mcphailb/puzzles/triangle.html

It puzzles me. I don't see how a triangle 2 units high and 5 units wide can have the same gradient as a triangle 3 units high and 8 units wide. Did they cheat?
 
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they aren't similar triangles. if they were you'd have a serious problem, they just appear close enugh for you to believe it when you look at it
 
recon said:
It puzzles me. I don't see how a triangle 2 units high and 5 units wide can have the same gradient as a triangle 3 units high and 8 units wide. Did they cheat?
Yes, they only appear to have the same gradient (as matt said). Try drawing the same problem, with great accuracy, on a piece of paper.
 

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