Commander

Well , at this stage I am working on my own in both Mathematical and Cosmology Fields to give wings to my own thoughts and not get restricted by existing Theories. Or else it will suppress my own line of thinking and intuitiveness. Sometimes I try and bring in my thinking in solving some Puzzles [where invariably heavy thinking is needed] into possible Generalization or Theories. For example We used a TRINARY SEARCH Algorithm in solving 39 Balls Problem better than a Binary Search Method. Similarly, here again the Property exhibited by Number 3 I noted from the Solution of another interesting Puzzle. We all know that 0 , 1 , 2 and 3 are the Bricks of Number Theories and 3 is on the Wall Dividing 0,1,2 and the rest 4.5.6 etc.. Like Fermat's theorem defines what is Possible with the Power 2 but not Possible for Powers 3 or more ! At this point I am looking at a possible link arising. Also if I add Walker's Equation too I can state : Every Possible Rational number can be expanded into a Sum of Powers of 2 Occurring Only Once and some Elements of the Matrix of Walker's Equation - perhaps occurring only once !

I think you misunderstood what I said. I know what Fermat's stated what you are quoting. That was his statement which needs to be proved or disproved.

Any Positive Natural Number N can be written as a positive sum of the Powers of 2 ! N = 2a + 2b ..... with as many terms as require where a , b , c etc are all Natural positive Numbers from [ 0,1,2,3 .....] These powers a, b, etc in the sequence will be PRESENT or ABSENT in the Equation only once. No repetition is required. This is nothing but a Binary Number representing the powers of 2 which add up to be equal to N. Similarly any N can be written as N = an additions or subtractions of Powers of 3 OCCURRING only once. N = 3a ± 3b ± 3c etc again a,b,c etc needing to appear only once ! Any number higher than 3 will not be able to produce such a Sum with their Powers Occurring only once ! Not possible to cover all N by 4a ± 4b ± 4c ....... or similarly by the powers of 5,6,7 ... or such higher numbers.

Thank you !

Thank you imatfaal !

imatfaal : Yes, your solution is correct and I have verified. I wish I had enlarged the text and read your second file earlier. I was busy creating the file / graphics to indicate my Solution. As I said earlier and as I see it now, there can be a few variations in the arrangement of weighings / balls. I CREDIT YOU WITH THE SOLUTION ! Well done !!

imatfaal : I had a look into the Book1.pdf and the second attempt. I have not yet convinced myself that there is a sure solution. I need to study more. However it was a brilliant attempt at it ! Thanks, yes it is indeed a very difficult problem based on very simple definition. I consider this as the best ever mechanical problem I have encountered and also a challenge to logic. I remember the days some years back I took a week's leave from Office to sit with the Rubic Cube and evolve a set of formulas and solution to Rubic cube without looking at the other published solutions and methods. It was a thrilling experience to derive a solution. I think it is now my duty to give the solution , ie my solution and I know there are some variations which can be also employed with balls arrangements to solve in a similar way. Can I now give my solution ? Regards

Hi imatfaal, Good you have found solution in 4 weighings ! I will study it and get back. Regards

But you already know that those 3 are either heavier or lighter so just 1 weighing will do !

If the above approach is taken : After the first two weighings we have B1 , B2, B3, B4, B5, B6, B7, B8, B9, B10, B11, B12, B13 within which the odd ball lies and we know whether it is lighter or heavier. sensei has solved it further in 3 more weighings by Binary Search Algo. Weigh B1 , B2, B3, B4, B5, B6 VS B7, B8, B9, B10, B11, B12 and if this balances it is B13 Else if it unbalances take the six in the correct pan [going up or down according to known condition of the oddball] and name it B1 , B2, B3, B4, B5, B6 On the 4th weighing B1 , B2, B3 VS B4, B5, B6 gets the oddball within three; name it B1 , B2, B3 On the 5th and last weighing B1 VS B2 and if this balances it is B3 or else B1 or B2 which tilts the scale in the known direction. imatfaal has pointed out that he can handle more number of balls at the 3rd weighing onwards using a different algorithm. Even if starting with 81 use the first two weighings to compare 27 balls instead of 13 and decide on 27 balls group within which lies the oddball with known deviational direction. Thereafter 9 VS 9 and if it balances the rest 9 or else among the 9 going the right direction. Its a kind of trinary algorithm. Then the 4th weighing reduces the oddball within 3 and thereafter it is the same procedure !

imatfaal : Sure, I'll wait and take your time. I know it is tough and those who persists with puzzle solving have great mental and analytical Energy ! Good Show and Keep it up ! Regards

Yes, that's the point !

TY Best regards Thomas PS : May not bother to correct old post as Moderators prefer not to encourage it ! Hi all, If BN is the number of balls which can be solved in this particular context of given problem [ie just one odd ball of different weight among B] in N number of weighings then, BN+1 = 3 ( BN + 1 ) If N = 4 and B4 = 39 then , B5 = 3 ( 39 + 1 ) = 120 The Solution for 39 Balls I mean the procedure I will give if there is a consensus ! I am sure imatfaal or sensei will find the solution on their own soon !! In the meantime I am looking for a demo Platform [ I have one for lesser number of balls but may not have the right to put it up here as I have not coded it myself ] Best Regards Thomas

Well I was in fact trying to create a puzzle for our local Research Lab Magazine and wanted to give it as Prove 1 = 22 + 23 + 24 ......... + 2n................. + 2∞ 32 + 33 + 34 ......... + 3n................. + 3∞ 42 + 43 + 24 ......... + 4n................. + 4∞ etc upto infinity downwards after I derived this as stated in my file. Then, many appreciated the beauty of the relation and I am in the process of finding whether anywhere this can be used !

Acme : I have this habit of emphasizing with Capitals now and then and please don't take offence to it. I know in internet parlance it also means Shouting and therefore I'll avoid it amap ! Regards Sensei : Yes, you have given the Solution in #17 to solve in 5 Steps. It is a nice Solution and I think you solve it always in 5 Steps and I can not see why you need a sixth step and in what sequence. Therefore you and imatfaal have solved it in 5 Steps ! Correct me and point out if I have missed out anyone else's Solution in 5 Steps. I am still to view Michel 123456's links and will comment on it if he/she wants. Regards Regards to all

Yes, of course it is in my head only right now ! If you give up and I have a consensus of demand for it I can reveal it !! Maybe next time you want me to pose a puzzle and give its solution in the Spoiler !!! Use your algorithm and give the solution here. In how many weighings are you solving it and what is the procedure ?

Can someone create a TEST PLATFORM for this Puzzle. You can pick and load Balls 1 to 39 on either side of the Balance. On the press of a button 'Weigh' the balance tilts or balances Of course everytime the Trial Starts one of the Balls is defined as Culprit randomly and oddity of Lightness or Heaviness too decided randomly. Just a kind of Demo / Work Bench

Yes, The Balls, Shapes , Coins or the Balance are not my Original Inventions ! I had already said there are similar problems I had solved earlier and they were not my own. ie with different number of balls. This problem of solving with 39 balls is my own proposal. If it was proposed by anyone already I am not aware of it. I too have solved a similar problem as early as 1973 and therefore I know similar problems have a long history of existence ! I have proposed the Puzzle with 39 Balls and another with 120 balls here which are MY OWN ! I also have a generic algorithm to calculate the maximum number of such balls that can be decided upon in N number of weighings ! WHICH IS ALSO MY OWN DERIVATION !! More Later ........

Well, I already said I can do it in 4 Weighings !

Sure, I hope so ! The doc asked me to undergo an operation to clear it up but I refused because it did appear that the Doc wanted the Money from it and also the improvement may not be permanent and therefore decided to BEAR THE SINUSITIS AS-IT-IS !!

Yes, its a Clue I can give that it can be done in 4 Weighings !

According to me we exist in a Space with Length Coordinates X , Y , X [the three dimensions which are least number to define any natural Volume Space] which can be used to designate any Point in the Entire Cosmos. We can fix the Origin of these three coordinates anywhere and calculate the distance of any other Point with respect to this origin.But as we are not able to make sure that this Origin is fixed and not moving in respect of rest of the Spaces, we are unable to do so. However our this INABILITY does not mean that we can not imagine a Reference Frame of Coordinates. Also, we can not declare that the Axes are NOT INFINITE just because we can not traverse them endlessly with ourselves or with an object with a Mass or A ray of Light or Particle or Even @ the Speed of Imagination. These coordinates are Endless and at the end of Space Occupied by some matter lies Limitless Nothingness or Empty Space. If the Physical Laws created by the Creator are such that matter Gravitates and condenses together ENDLESSLY without any Limit THE ENTIRE MATTER IN THE COSMOS are related by Mutual Forces and therefore there can be only one Universe. But, if there is a Limit to the amount of matter which can congregate together [Like the Limit which produces a star or Black Hole or Big Bang] then there is a possibility of many such Universes like our Own Earth-Universe in the Cosmos. Whatever be the case BEYOND ALL THESE BOUNDS will lie rest of the Nothingness and Empty Spaces. Therefore we need to understand exactly what happens with the Gravitational Forces and Behaviour of Matter as a Consequence. TIME is one of the Dimensions adopted by us to evaluate the Changes in a Body and Relative Change wrt other Bodies. The various Forces and Factors such as Mass, Weight , Temperature, Pressure and Fields etc gives raise to other dimensions too and when we handle Multidimensions to Observe and Evaluate changes we adopt higher Mathematical Functions and Complicated Theories. But TIME in itself has been defined as a delay segment between two Events or States of a Body. Except in the ABSOLUTE EMPTINESS or NIL-SPACE there are always CHANGES effected on any Point in Space or Body and its State and TIME ELAPSED becomes a very important Parameter. Of course there is an ABSOLUTE TIME REFERENCE which too is only Imaginary and NOT REFERABLE. We need to bank on possible Relative References to Fix the Current Time, Measure the Lapse of Time and Use our Measurement in the Equations to solve Problems and define States and Events.

I meant BN to mean the number of Balls which can be solved in N Weighings. For example I can solve 120 Balls in 5 Weighings which means B5 = 120 ! That means if the given number of Balls are 120 instead of 39 the answer will be : 5 Weighings and the procedure to find that Culprit Ball in those 5 Weighings !! That can be taken as a second Puzzle !!!

Acme : Thanks !

Hi, Thanks Yes, all balls need to be identical in every aspect st an wisecrack does not use visible cues ! Can be Cubes or any other shape too - just equal is weight except just in one case and ONLY Weight IS THE CRITERIA TO USE and DEDUCE FROM !! Hi, Can you suggest a generic algorithm. I think I have one which gives BN which means the number of Balls that can be decided upon [spot and Fix the Direction of Variation Lighter or Heavier]as a Function of BN-1 PS : I read your Spoiler later and I think your Solution is similar to that of imatfaal and may produce a Solution in 5 Turns ! Good Try !!