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Explanation by Julian H. Stacey

The Puzzle

Search on the net for your own copy of one of these two graphics files (which look the same):
  • Name: 1-44.gif
    • Size in bytes: 19476
    • Md5: 66c33b53f897a56497ef19c38795f1ed
  • Name: triangle.gif
    • Size in bytes: 22241
    • Md5: 59761ba4d4e33675f5dac9ce909838f1
They are not online here in case they are copyright & not public domain, though I received mine as semi infinitely forwarded public joke type mails, & "strings 1-44.gif | grep -i copyright" & strings 1-44.gif | grep -i copyright "strings triangle.gif | grep -i copyright" shows nothing.

The Explanation

Best enjoy solving it yourself I suggest.

But if you want the answer, scroll down.








































  It's an an optical illusion.

The Illusion

  • One feels the diagonal looks the same on both the red & green triangle.
  • One assumes a straight line joining red & green diagonals.
  • One assumes same straight diagonal line on both diagrams.
  • One knows 2 straight lines at same angle must cover same area & number of squares underneath.

The Reality

  • There is not one diagonal on each diagram, but two.
  • The diagonals are different but optically similar gradients:
    • Red:......: 3 vertical in 8 horizontal = 0.375 = Shallower slope
    • Dark Green: 2 vertical in 5 horizontal = 0.400 = Steeper slope
  • The top diagram is thus somewhat concave, or hollow ( think of a very large white / invisible squashy ball on top left, pushing right, into the dent left by the 2 coloured slopes.
  • The lower diagram is opposite: somewhat convex, it bulges upwards in the middle
  • The lower diagram, as it has bulged outward, has used all its material up doing so, & has left a space underneath.

Prior email answer to a friend (quoting dc (a Unix tool)).

Did you know the attached puzzle?
No, & I must admit it took me a few minutes to spot the answer. Mainly 'cos I knew it must be damn simple & I should be spotting it pretty much instantly. I was on the right lines when I guessed diagonals probably weren't parallel, but instead of peering closer, or just sitting back & thinking, I took to counting squares on X & Y, which I guess most do. Damn I must be getting slow. :-) Anyway the Unix based proof of the differential gradient & visual trickery is below:
dc 200000 5 / p 40000
dc 300000 8 / p 37500


(Puzzles may be fun occasionally, but ... ) I prefer real life design, debug & maintenance problems an Internet & Systems Engineering Consultant is paid to solve. Please contact me if you know of a project, Thanks.



A little integer arithmetic tool I think has been in every Unix since or before 1977. It works in "Reverse Polish" notation ( if you stack numbers & operators (ie + - / * ) in right order, you don't need any brackets on equations, & that cut calculator electronics cost, or bought you a better calculator for the same money, or an affordable calculator instead of no calculator, back when calculators were expensive (even now it saves space for calculator brackets keys).


A Free & better operating system that precedes & surpasses viral Windows My favourite version FreeBSD had 125650 Free packages at March 2005. (Example of calculators available to FreeBSD include eg hexcalc in the math collection & xcalc (unlisted in but part of xorg-clients in x11, (along with XFree86-4-clients & XFree86-contrib & probably also KDE etc )

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