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October-November 2009, Puzzle nr 123
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Send to a Friend Puzzle # 123
Z puzzle

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The Mark of Zorro
   With just 3 straight lines through the zed above (fig. 1) form the LARGEST possible number of triangles (see example). You must also prove that your solution is the best. Hasta luego!
Just for an anecdote, few people know that Guy Williams, the best Zorro ever, was fond of mathematics, chess and astronomy...

Difficulty level: bulbbulb, basic geometry knowledge.
Category: dividing-the-plane puzzle.
Keywords: triangles, line segments.
Related puzzles:
- Red monad,
- Stairs to square.

Source of the puzzle:
© G. Sarcone.
You cannot reproduce any part of this page without prior written permission.
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solution 1

As you can observe in the fig. 3 above, for obtaining the greatest number of regions when dividing a surface with straight lines each line segment MUST intersect ALL the other ones, and by any intersection point should pass ONLY two lines. The example in fig. 3C meets all the criteria we have outlined, thus it contains the greatest number possible of regions: 7 regions instead of 6.

Applying to our problem the empirical criteria above, we obtain the following diagram:

Solution 2

Therefore, the maximum number of triangles obtainable by intersecting the large Z with 3 segment lines is SEVEN.
Notice that each of the 3 segment lines touches the two others + the diagonal and both parallel lines forming the capital letter Z.

cup winnerThe Winners of the Puzzle of the Month are:
Serhat Duran, Turkey Flag turkey - J. R. Odbert, USA USA flag - Vishal Dixit, India indian flag - Congratulations!

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Math fact behind the puzzle
crescent like figureCalling the number of line segments n, it can be demonstrated that the maximum number of portions a convex polygon can be divided with them is:
   (n2 + n + 2)/2 [see fig. 3 above]
but in the case of a concave polygon such as a crescent-like figure (see fig. 5) it will be:
+ 3n + 2)/2
The numbers generated by the first formula (e.g. 2, 4, 7, 11, 16, 22, ...) are called “Pizza numbers”.


© 2006 G. Sarcone,
You can re-use content from Archimedes’ Lab on the ONLY condition that you provide credit to the authors (© G. Sarcone and/or M.-J. Waeber) and a link back to our site. You CANNOT reproduce the content of this page for commercial purposes.

You're encouraged to expand and/or improve this article. Send your comments, feedback or suggestions to Gianni A. Sarcone. Thanks!
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