Solution to SrImati rAjammA's problem
As I mentioned in my hint, the original motif redrawn on a hexagonal grid, was placed at the centre (in red) and around it six (in dark blue and light blue) more were arranged so that they are all in close contact. Around this second layer, 12 more (in purple and pink) were fit. Six more (in orange) were placed in the fourth layer. We have thus 1+6+12+6 = 25 basic motifs. Actually the fourth layer is not full. There must be 18 in the fourth layer. In the figure on the right side, I filled all the 18 (in orange, red and rose). This is more full without leaving any vacancy. In this there are 1+6+12+18 = 37 basic motifs. If the next layer is similarly filled, there will be 24 in that. The over all basic symmetry is only three-fold. This type of exercise educates one as how to fill motifs and create a lattice with translational symmetry. Crystals are filled with such molecules infinitely along the three directions in three dimensional space. Enjoy!
Regards! - mOhana
Comments
dibbutn
Mon, 2012-06-25 22:01
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Congrats Sir for giving the right answer to mam's puzzle... looks really pretty
anirudh
Mon, 2012-06-25 22:26
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Great Sir, thanks for sharing the logic and calculations
Suguna Murugesan
Mon, 2012-06-25 22:32
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congrats sir .... thanks for sharing explanation and giving the full version
smahalakshmi
Mon, 2012-06-25 22:36
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Wow, superb one, JKM sir. You have given 2 answers for the puzzle. Both the designs looks beautiful. Also enjoyed reading your basis of arriving at this solution.
jayamohan
Mon, 2012-06-25 23:29
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JKM sir, you have clearly explained the method to proceed from the center and arrive at the correct solution!
chandhini.selvaraj
Tue, 2012-06-26 00:06
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thanks for your guidance uncle
awesome creation! thanks for your description uncle 
vidhyaprasu
Tue, 2012-06-26 01:53
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Excellent sir, Beautiful explanation for the solution....Thank you sir
rajamma_2
Tue, 2012-06-26 05:00
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Superb. Both the outcome are the real perfect solutions. And your explanation will give the members to invent and also solve such puzzles.Thank you Sir for taking such interest and encouraging and educating us. In the first two puzzles I wanted to mention that members should go thru your gallery to get some idea or guidance to connect similar patterns, thereby craeting different design in the gap!
Can I request you to give some work like this to freshen up our brains?
But the response was poor. So I left it. But I wrote that for mOhanaji these type of puzzles will be "Jujoobee".
Sir,
vijaysowmya
Tue, 2012-06-26 05:24
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Wow...Excellent creations Sir. Thanks for your explanation too as now I am able to understand clearly the logic.
ammuchandhini
Tue, 2012-06-26 18:04
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Wow....two answers for this puzzle when we were pulling our hair for atleast one possible correct answer jkm sir....
such lovely creations and thanks a ton for d description too 
subashini
Tue, 2012-06-26 20:54
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excellent and beautiful work sir.Thanks for sharing.
lakshmiraghu
Thu, 2012-06-28 06:10
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Excellent creation..., Beautiful explanation