## Lethal Inspection

This post is motivated by the New York Times NumberPlay related to the Mathematical Secrets of the Simpsons.

I , sadly, must confess that despite having identified the “Alien Script” as a cipher and knowing how to approach inspiration was lacking and I needed help to solve it.

It did, however, afford me the opportunity to play with Mathematica to assist me. Spoiler alert the followign gif decodes the cipher. The code used for the interface follows. Although it appears daunting it is quite compact and the majority of the length relates to the image. YOu can copy and paste into a Mathematica notebook.

DynamicModule[{uv, rule}, uv = Table[Unique[], {Length[lett]}]; rule = Table[ Row[MapThread[ Row[{Style[#1, Red, Bold, 16], " ", PopupMenu[Dynamic[#2], Prepend[CharacterRange["a", "z"], #1], MenuStyle -> {Bold, 16}]}] &, {lett, uv}][[j]]], {j, {1 ;; 4, 5 ;; 8, 9 ;; 12, 13 ;; 16, {17}}}]; Dynamic@Row[{Column@rule, Framed[Image[img, ImageSize -> {300, 180}]], Column[{Style[str, 16], Style[StringReplace[str, Thread[lett -> uv]], Red, 16]}, Dividers -> All]}, Frame -> True, FrameStyle -> Blue], Initialization -> (str = "FDDO DGYPT ZTLX SDHY OMKF JMEP MHO EFKTFYDO XESTFL KD CTJ YMC OMHHTP"; lett = Union[Join @@ (Characters /@ StringSplit[str])]; img = \!$$\* GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJzsnQm8VtP3/5GQylAqoQyRCiFTpohEhYgGQyVSURqQSkqiSBENFFJESkoR yVChQhS+macUZebrO/1U9z77//4/n9dZr3XPc+9zb7cRZ71e3dfTec6zzz5r r/VZn7X2Pvvsf3m3Zh2222abba7biT/N2vWqf+217fpcsD3/Oa3P9VdeXoIP P/Lv62232eb/fw6JJJJIIokkkkgiiSSSSCKJJJJIIokkkkgiiSSSSCKJJJJI 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Categories: Uncategorized

## Love and Math

December 15, 2013 1 comment

This book is both an invitation to dive into Mathematics and the account of personal journey  from humble beginnings and in the face of structural prejudice. Mathematics had a transforming role as a universal, objective yet dynamic discipline.  Professor Frenkel, though clearly an exceptional Mathematician, faced major obstacles to pursuing his passion and  profession.   The book  provides insights into the structural ant-semitism in Russia and the clandestine efforts of people of good will  to fight against this.

The Mathematical story is weaved through this personal story. This certainly has motivated me to learn about the Langland’s programme and modern Mathematics.  I look forward to diving into this, if and when I get time. The book provides a taste focused on concepts.

I think the journey through adversity and making difficult decisions along the way  was poignant and inspirational.  The pursuit of Mathematics, to discover  the hidden deep connections of the Mathematical universe and how it relates to the remarkably comprehensible “real world”, was its own reward and a lingua franca bringing together people of diverse origins, experience and passions..  Professor Frenkel appears to be  a Platonist, arguing an independent universe of Mathematical Truth that humans discover (and do not create)…I think I am to but this is a deep philosophical question and I have only an extremely limited field of view. This book encourages me to broaden it.

Categories: books, Mathematics

## Roll the Dice

Some musings from the NY Times NumberPlay here. The   first game is fair but needs a lot of capital and the second is maniestly unfair. The following are 100 simulations of 10000 plays:   An alternative offered up from a comment: Tellyouwhat,I′llgiveyouadiscounttomakeiteven.Insteadofrisking18, if you give me 15, I′ll let you roll six times,and I′ll pay you 12 for each 6 you roll.” is also unfair: for every 15 dollars paid the expected return is 12 dollars. 10000 siumulations shows net gain (mean -3):

Categories: Uncategorized

## The Menace Strikes Back

The Phantom Menace has returned a little early this year. Tens of kilograms of material to gather every few days. As if a message, the solitary piece of fruit on the car roof symbolizes the Menace’s stubborn persistemce…but it will tire.

My counting exercise (missed some and overcounted others and not the complete area), tells me I have over 500 pieces of fruit to gather…a prolific Phantom this year…

Categories: Uncategorized

## Dividing the Spoils

I confess to taking an embarrassingly long time to understand thisNew York Times NumberPlay.

Look at the image before clicking the hyperlink. Eventually, I appreciate all of the “gold” segments add to 4/n with first row representing n=2. The other segmentation allows the same partitioning for another n-1 people.

How to partition? What rules? Note partitioning $\frac{4}{n}=\sum^k_{j-1}\frac{1}{x_j}$ is note unique, e.g. 4/5 =1/2 +1/5+1/10 or 4/5=1/2+1/4+1/20…

After some thought, and more play I found an algorithmic way to reproduce the first terms of the sequence presented. This does not guarantee this is the general solution. I certainly have not worked out the choice of what gold bar to colour. I very much look forward to the answer.

The following is my code and an extension of the result.

g[n_, d_] := Module[{num, sc, den, div, sel2,}, If[Numerator[n/d] == 1 || IntegerQ[n/d], Return[{n/d}]]; num = Ceiling[d, n]; sc = num/n; den = sc d; div = Divisors[den]; sel2 = DeleteDuplicates[ Select[Tuples[div, 2], Total@# == num &], #1 == Reverse@#2 &]; If[sel2 == {}, Return[First@ Union[Sort /@ Select[Tuples[div, 4], Total@# == num &]]/den] ]; First@SortBy[sel2, Abs[#[[1]] - #[[2]]] &]/den] 

2 {2}
3 {1/3,1}
4 {1}
5 {1/10,1/5,1/2}
6 {1/3,1/3}
7 {1/14,1/2}
8 {1/2}
9 {1/9,1/3}
10 {1/5,1/5}
11 {1/33,1/3}
12 {1/3}
13 {1/52,1/26,1/4}
14 {1/7,1/7}
15 {1/10,1/6}
16 {1/4}
17 First[{}]/85
18 {1/9,1/9}
19 {1/95,1/5}
20 {1/5}

For 4/17 there are no sums of 2 or 3 with the method used, however: 4/17=1/5+1/85+1/85+1/85…

UPDATE
After feedback from NumberPlay some refinement, to find the smallest partition of 4/n with the largest smallest piece:

 fun[n_, d_, num_, m_] := Module[{lcm, div, sc}, lcm = m LCM[n, d]; div = Tuples[Divisors[lcm], num]; sc = lcm/d; Union[Sort /@ Select[div, Total@# == sc n &]/(lcm)]] gf[n_, d_] := Quiet@If[Numerator[n/d] == 1 || IntegerQ[n/d], {n/d}, If[func[10, n, d, 2] === Last[{}], func[10, n, d, 3], func[10, n, d, 2]]] 
I have not tidied this up or thought sufficiently deeply to show how to limit search space…time does not permit at present but it seems I am on a better track and I look forward to the solution.
 Grid[Table[{j, Style[gf[4, j], 20]}, {j, 2, 20}]] 
2 {2}
3 {1/3,1}
4 {1}
5 {1/10,1/5,1/2}
6 {1/3,1/3}
7 {1/14,1/2}
8 {1/2}
9 {1/9,1/3}
10 {1/5,1/5}
11 {1/33,1/3}
12 {1/3}
13 {1/52,1/26,1/4}
14 {1/7,1/7}
15 {1/10,1/6}
16 {1/4}
17 {1/102,1/17,1/6}
18 {1/9,1/9}
19 {1/95,1/5}
20 {1/5}

Categories: Mathematica

## Pushing Buttons

This post is motivated by the New York Times Numberplay puzzle: Alex and the Button Lock
This can be approached in a number of ways. The answer is a De Bruijn sequence and although this could be solved with in-built commands in Mathematica and some further manipulation, I decided to create my own graph and play.

The two button lock can be solved by a sequence of 5 pushes, e.g. ABBAA or AABBA.
The three button case has 27 triples that must be coded in minimum sequence length.
In the following, an edge exists after adding a button push, viz. adding A to AAA leads to subsequence AAA, AAA->AAA (self loop), or adding B to AAA leads to subsequence AAB,AAA->AAB and so on.
The minimum sequence length with all the triples is a Hamiltonian cycle.

The following graphics illustrate: the underlying graph, a Hamiltonian cycle, “untangled” cycle and an animated gif of the untangling (apologies for the slowness and errors).

One solution (this can obviously be cycled and I have not explored other Hamiltonian cycles).

Hamiltonian Cycle

“Untangled”

Animated gif (“warts and all”):

One solution (therefore):

AAABAACABBBACBABCACCCBBCBCCAA

Some of the code used:

Setting up underlying graoh
 gen = {"A", "B", "C"}; tup = Tuples[gen, 3]; ge = Map[Function[x, {#, Take[Join[#, {x}], -3]}], gen] & /@ tup; gr = DirectedEdge @@@ Join @@ Map[StringJoin, ge, {3}]; grlist = Join @@ Map[StringJoin, ge, {3}]; grp = Graph[gr, VertexLabels -> Placed["Name", Center], VertexSize -> 1, GraphLayout -> "SpringEmbedding", VertexLabelStyle -> Directive[White, Bold, FontFamily -> "Kartika", 16], VertexStyle -> Red] 
Finding and rendering solution:
 hc = FindHamiltonianCycle[grp] HighlightGraph[grp, First@hc, GraphHighlightStyle -> "Thick"] hcl = hc[[1]][[All, 1]]; ans = StringJoin @@ {First@hcl}~Join~(StringTake[#, -1] & /@ Rest@hcl) 

Graphical interface to untangle the Hamiltonian cycle
 coord = PropertyValue[{grp, #}, VertexCoordinates] & /@ (StringJoin /@ tup); DynamicModule[{pt = coord}, LocatorPane[Dynamic[pt], Dynamic[r = Thread[StringJoin /@ tup -> pt]; Graphics[{{Blue, Arrowheads[{0, 0.05, 0}], Arrow /@ (grlist /. r)}, {Red, Thick, Arrowheads[{0, 0.05, 0}], Arrow /@ (Partition[hcl, 2, 1] /. r)}, {Text[ Framed[#, Background -> White], # /. r] & /@ (StringJoin /@ tup)}}]], Appearance -> None]] 

There are no doubt better ways to do this. I also not I display path (not cycle) as this is the solution.

Categories: Uncategorized

## Triangles, Circles and Spheres

The ‘real world has provided precious few opportunities for musings. The following is motivated by a question on Mathematica Stackexchange.

The following code aims to find the circumcentre of a triangle defined by 3 points in $R^3$.
 cc[{p_, q_, r_}] := Module[ {v1, v2, b1, b2, norm, n1, n2, cntr, rad, un}, v1 = Normalize@(q - p); v2 = Normalize@(r - p); b1 = (p + q)/2; b2 = (r + p)/2; norm = Cross[v1, v2]; n1 = Normalize@Cross[norm, v1]; n2 = Normalize@Cross[norm, v2]; cntr = b1 + s n1 /. First@Solve[b1 + s n1 == b2 + t n2, {s, t}]; rad = Norm[p - cntr]; un = Normalize@Cross[norm, v1]; {cntr, rad, v1, un} ]

 

plcc[{p_, q_, r_}] := Module[{ctr, radius, un1, un2, p1, p2}, {ctr, radius, un1, un2} = cc[{p, q, r}]; p1 = Graphics3D[{{Black, PointSize[Large], Point@{p, q, r}}, {Polygon[{p, q, r}]}, {Red, PointSize[Large], Point@ctr}, {Opacity[0.3], Sphere[ctr, radius]}}]; p2 = ParametricPlot3D[ ctr + radius Cos[t] un1 + radius Sin[t] un2, {t, 0, 2 Pi}]; Show[p1, p2, Axes -> True]] 

Testing on 100 random triples:

Peace to all…

Categories: Mathematica, Mathematics