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[[File:Missing Square Animation.gif|thumb|right|300px|Missing square puzzle animation]]
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
The '''missing square puzzle''' is an [[optical illusion]] used in [[mathematics]] classes to help students reason about geometrical figures. It depicts two arrangements made of similar shapes in slightly different configurations. Each apparently forms a 13×5 right-angled [[triangle]], but one has a 1×1 hole in it.


==Solution==
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
[[File:Missing square puzzle.svg|thumb|right|200px|The missing square shown in the lower triangle, where both triangles are in a perfect grid]]
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The key to the puzzle is the fact that neither of the 13×5 "triangles" is truly a triangle, because what appears to be the [[hypotenuse]] is bent. In other words, the "hypotenuse" does not maintain a consistent [[slope]], even though it may appear that way to the human eye. A true 13×5 triangle cannot be created from the given component parts. The four figures (the yellow, red, blue and green shapes) total 32 units of area. The apparent triangles formed from the figures are 13 units wide and 5 units tall, so it appears that the area should be <math>\textstyle{S=\frac{13 \times 5}{2}=32.5}</math> units. However, the blue triangle has a ratio of 5:2 (=2.5:1), while the red triangle has the ratio 8:3 (≈2.667:1), so the apparent combined [[hypotenuse]] in each figure is actually bent. So with the bent hypotenuse, the first figure actually occupies a combined 32 units, while the second figure occupies 33, including the "missing" square. The amount of bending is approximately 1/28th of a unit (1.245364267°), which is difficult to see on the diagram of this puzzle. Note the grid point where the red and blue triangles in the lower image meet (5 squares to the right and two units up from the lower left corner of the combined figure), and compare it to the same point on the other figure; the edge is slightly under the mark in the upper image, but goes through it in the lower. Overlaying the hypotenuses from both figures results in a very thin [[parallelogram]] with an area of exactly one grid square—the same area "missing" from the second figure.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


===Principle===
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According to [[Martin Gardner]],<ref>
{{cite book
|last= Martin
|first= Gardner
|title= Mathematics Magic and Mystery
|year= 1956
|publisher= Dover
|pages= 139–150
|isbn= 9780486203355
}}</ref> this particular puzzle was invented by a [[New York City]] amateur magician, [[Paul Curry]], in 1953. However, the principle of a dissection paradox has been known since the start of the 16th century. The integer dimensions of the parts of the puzzle (2, 3, 5, 8, 13) are successive [[Fibonacci numbers]]. Many other geometric [[dissection puzzle]]s are based on a few simple properties of the Fibonacci sequence.<ref>{{cite web |publisher=Math World |last=Weisstein |first=Eric |title=Cassini's Identity |url=http://mathworld.wolfram.com/CassinisIdentity.html}}</ref>


[[File:Missing square puzzle dimensions.png|thumb|right|200px|Missing square puzzle dimensions]]
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


==Similar puzzles==
<!--'''PNG'''  (currently default in production)
[[File:Loyd64-65-dis b.svg|thumb|right|200px|[[Sam Loyd]]'s paradoxical dissection]]
:<math forcemathmode="png">E=mc^2</math>
[[Sam Loyd]]'s paradoxical dissection. In the "larger" rearrangement, the gaps between the figures have a combined unit square more area than their square gaps counterparts, creating an illusion that the figures there take up more space than those in the square figure. In the "smaller" rearrangement, each quadrilateral needs to overlap the triangle by an area of half a unit for its top/bottom edge to align with a grid line.


[[File:Missing square edit.gif|thumb|left|150px|A variant of Mitsunobu Matsuyama's "Paradox"]]
'''source'''
Mitsunobu Matsuyama's "Paradox" uses four congruent [[quadrilateral]]s and a small square, which form a larger square. When the quadrilaterals are rotated about their centers they fill the space of the small square, although the total area of the figure seems unchanged. The apparent paradox is explained by the fact that the side of the new large square is a little smaller than the original one. If ''a'' is the side of the large square and ''θ'' is the angle between two opposing sides in each quadrilateral, then the quotient between the two areas is given by sec<sup>2</sup>''θ'' − 1. For ''θ'' = 5°, this is approximately 1.00765, which corresponds to a difference of about 0.8%.
:<math forcemathmode="source">E=mc^2</math> -->


{{-}}
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


==References==
==Demos==
{{Reflist}}


==External links==
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
{{Commons category|Missing square puzzle}}
*A printable [http://www.archimedes-lab.org/workshop13skulls.html Missing Square variant] with a video demonstration.
*[http://www.cut-the-knot.org/Curriculum/Fallacies/CurryParadox.shtml Curry's Paradox: How Is It Possible?] at [[cut-the-knot]]
*[http://www.archimedes-lab.org/page3b.html Triangles and Paradoxes] at [[archimedes-lab.org]]
*[http://www.marktaw.com/blog/TheTriangleProblem.html The Triangle Problem or What's Wrong with the Obvious Truth]
*[http://www.mathematik.uni-bielefeld.de/~sillke/PUZZLES/jigsaw-paradox.html Jigsaw Paradox]
*[http://www.slideshare.net/sualeh/the-eleven-holes-puzzle The Eleven Holes Puzzle]
*[http://www.excelhero.com/blog/2010/09/excel-optical-illusions-week-30.html Very nice animated Excel workbook of the Missing Square Puzzle]
*A video explaining [http://www.youtube.com/watch?v=eFw0878Ig-A&feature=related Curry's Paradox and Area] by James Stanton


{{DEFAULTSORT:Missing Square Puzzle}}
 
[[Category:Optical illusions]]
* accessibility:
[[Category:Fibonacci numbers]]
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
[[Category:Elementary mathematics]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
[[Category:Mathematics paradoxes]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
[[Category:Puzzles]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
[[Category:Geometric dissection]]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.
 
==Test pages ==
 
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]
 
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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