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The Toy

Opineddu, the Cat and the Fox had just had dinner and, as they weren't hungry at all they had a light one : the Fox had a large boiled fowl and the Cat about six pounds of fried bonnetmouth. 'See Opineddu, the Fox starts to brag, how the time has changed us. Long ago I fell down and since then I'm a poor lame. Our fella the Cat has been blind since he was born. We haven't got any luck,we're poor but honest men. We're not like our grandfathers that had stolen your late lamented grandfather's gold. We're intelligence entrepreneurs! We invest only on bright ideas and just now we've got one that suits you well. By the way, how much has Pappafogu given to you? Sixty four gold Napoleons? It's on the upand up. Wait...I fetch my walking stick and come to illustrate our plan to you. Well, you put the money and we put the labour. It's a continental plan, scientific stuff, pure maths!
We're going to show that 64 is equal to 65. On this account we built a chessboard as you can see from this design. Where does the labour lie? It suffices to put the 64 gold Napoleons, one for each small square of the square chessboard. In a minute we disassemble and reconstruct it forming a rectangular chessboard with the various parts of the square exactly like they were. Don't you realize anything Opineddu? The square is divided into two congruent right angled triangles, their catheti measuring 8 and 3 and into two congruent trapezia with their major bases measuring 5 and the minor ones 3. We will reconstruct those same figures without no trick nor deceit to form the rectangle and there aren't any remains. Right? If you count well there is one more small square in the rectangle and therefore one more gold Napoleon. It suffices to do this operation sixty four times a day and we'll double our capital within a year. We'll be millionaires, say, not millionaires but billionaires! This plan is called ' su jogu',- the play-, and it was studied by my fella the Cat: he is a Siamese cat of course, and it is all written in Sardinian language to be clearly understood without racking your brain in doing that. Have a look at it and then you'll tell me what you think about it'. Opineddu was taken aback. The same,the splitted figures, but for a small square more: a Napoleon more! In no time they were going to be real capitalists and, last but not least, together with so nice people! But as soon as he pulled his wallet out of is pocket to set the Napoleons on the chessboard, the Cat and the Fox vanished in the air together with their good intents and the mug's gold coins. Fortunately we have the plan left and will explain it making a detailed analysis in Sardinian. In 'su jogu' expression we can note that the trickworks when the arguments are consecutive terms ofthe Fibonacci's series (we refer the interestedreader to another hard literary toil entitled Fibonacci).Without going into tiring technical details and not meaningful at all at the moment, we declare that the greater the index of the Fibonacci's series term we take into account the more, simplifying the expression, an infinitesimal term appears that measures the area of the small square miraculously appeared. All that means that for small numbers the squareand the splitting done by the Fox is not the same of the rectangular reconstruction. But forlarge,really large numbers, such inequality disappears and things fit into place again. With Opineddu scorn and ridicule.

Clear[f,n]
Expand[
Simplify[
Divide[
Expand[
(f[n]+f[n-1]) f[n-1]-f[n]^2-1],-f[n-1]^2]]]/.
{f[n]^2/f[n-1]^2->N[GoldenRatio^2],
f[n]/f[n-1]->N[GoldenRatio]}

-2
0. + f[-1 + n]

"Since our Regional Council has started pine reforestation there has been a home upsurge in the Opineddos * growing. But they weren't lacking in the past too!

by Uncle Marthine and the IGEA 3rd class of Macomer Commercial School.

*Opineddos=small pines.

And this is the Fox and the Cat's plan

(*everything we write into a parenthesis and mark with an asterisk is ignored by the program*)

sujogu[j_,h_,k_]:=

(*sujogu[j,h,k],a square and a rectangle are drawn and look like as they were done by the same geometrical figures: two triangles and two trapezia but with different areas.j,h,k are three natural numbers that belong to the Fibonacci's series. If you wish to get through with that trick you'll have to use other types of numbers*)

(*Module[{var}] it obliges the variables that are its own arguments to be internal to the program itself*)

pq=Table[N[x+I y],
{x,0,k},{y,0,k}];

(*pq is the short form of some points of the square. It consists in a set of complex numbers of the a+bI form.
Table is an iterator, x and y its variables in the variation field of a lower nought extreme and upper k extreme.*)

(*coordquad is the coordinates in the square. It is a set of number couples on the Cartesian plane that are the coordinates of the points where horizontal and vertical lines meet*)
(*Map[arg] is a function which is applied to a list of numbers or everything else you want and on the occasion it separates real parts from imaginary ones of numbers of the a+bI form to do a couple of the <a,b> form*) (*#, number, and &, ampersand, are the Function[arg] short form or rather we can say it is the postfixed notation of Function*)

(*orquad means horizontals of the square. Line, out of a graphic context, is a function that in a formal way joins different  points with linear parts. On this occasion the coordquad numeric couples represent coordinates of points on the Cartesian plane that are related among them in a way that, from one point to the other, a horizontal line goes through. Map arranges that Line is applied to eachone of the coordquad couples*)

(*verquad is a short form for verticals of the square. The coordinates of a point, to be able to draw vertical lines, have to exchange x that has to became y and vice versa. In the set of numbers this task is carried out by the Transpose[arg] function*)

Graphics[
AspectRatio->Automatic,
DisplayFunction->Identity];

(*quadratu -square- is the name we give to a graphic object drawn with the horizontal and vertical lines orquad and verquad. DisplayFunction->Identity prevents the program doing the graphic output. Furthermore, when we need the graphic we'll remove this key*)

pr=Table[N[x+I y],
{x,k+2,k+h+2},{y,0,k+h}];

(*pr is for some points of the rectangle*)

coordrett=Map[{Re[#],Im[#]}&,pr,{2}];

(*coordrett is for rectangle coordinates*)

orizzrett=Map[Line,coordrett];

(*orizzrett is for the rectangle horizontals *)

verrett=Map[Line,Transpose[coordrett]];

(*verrett is for the rectangle verticals*)

rettangulu=Show[
Graphics[
Join[orizzrett,verrett]],
AspectRatio->Automatic,
DisplayFunction->Identity];

(*rettangulu -rectangle- is the name of a graphic object drawn by means of vertical and horizontal lines.
AspectRatio->Automatic is an instruction that arranges that right angles are properly right angles*)

l1=Show[
Graphics[
{RGBColor[1,0,0],
Thickness[0.008],
Line[{{0,0},
{k,0},
{k,k},
{0,k},
{0,0},
{k,j},
{0,j}}]}],
DisplayFunction->Identity];

(*l1 is a line that joins the points in a sequence and their coordinates are the subject matter of Line. RGBColor[arg1,arg2,arg3] is a graphic instruction for the red colour, in this case*)

l2=Show[
Graphics[
{RGBColor[1,0,0],
Thickness[0.008],
Line[{{j,j},
{h,k}}]}],
DisplayFunction->Identity];

(*l2 is another line.  Thickness[val] is an instruction that regulates the line thickness*)

l3=Show[
Graphics[
{RGBColor[1,0,0],
Thickness[0.008],
Line[{{k+2,0},
{h+k+2,0},
{h+k+2,h+k},
{k+2,h+k},
{k+2,0},
{h+k+2,h+k}}]}],
DisplayFunction->Identity];

(*l3 is another line. The object of the Line function are linked to the variables j,h,k we selected*)

l4=Show[
Graphics[
{RGBColor[1,0,0],
Thickness[0.008],
Line[{{k+2,k},
{k+2+j,k},
{k+2+h-j,h},
{k+2+h,h}}]}],
DisplayFunction->Identity];

(*l4 is another line*)

DisplayFunction->\$DisplayFunction];];

(*the whole work is displayed by means of the Show function with which the DisplayFunction key is stopped.
The program works inserting the word su jogu and giving it proper subjects*)

sujogu[3,5,8]

sujogu[8,5,8];

The duel

"Once upon a time there were three gentlemen: Antiogu, Bartolu and Cosomeddu who had had words and had challenged themselves to a duel.

A duel for three: we could say a 'threeuel',a pistol fighting, one for each one of them with a single shot. Only one of them could shoot in turns. So, he who shots, shots and he does not, receives the shot. Call A,B and C the three duellers. We know that when A shoots, he hits the target once out of three, B twice out of three and C nerver fails a shot. In probability terms: P(A)=1/3,P(B)=2/3,P(C)=1.
Now the duellers have to establish who is the first one to shoot, who the second one and who the last one. They reject the idea of drawing by lot using some pieces of paper and so they start to discuss about it and, in a democratic way, they agree that the shooting order has be established by vote. Once fixed the order, each one of them will be able to shoot at him who he wishes. If the three duellers have discussed in a proper manly way as to obtain the utmost personal satisfaction (we would say to escape death and to cause their opponents the greatest harm they could), how will the story end?
Antiogu thinks like that: ' If I first shoot at Bartolu and kill him, Cosomeddu will shoot at me and, as he never fails, will kill me. It doesn't suit me. If, on the contrary, I shoot at Cosomeddu first and kill him, Bartolu will shoot at me and, as he is a good shot,he migth kill me with a 2/3 probability.If I shoot at Bartolu first and I don't kill him, he'll no more shoot at me because I'm disarmed and then he'll try to shoot at Cosomeddu, otherwise the two of us are dead. If Bartolu kills Cosomeddu, I'll be safe otherwise with a 1/2 probability Cosomeddu will shoot at me. If I pretend to shoot at Cosomeddu first and I don't hurt him, I'll be disarmed and I no more be a danger for them and so Cosomeddu, if he is the second one to shoot, will shoot at Bartolu for sure, otherwise he'll have a 2/3 probability to be a dead man and I'll be certain to be a safe man. I need to shoot a high shot and the second one has to be Cosomeddu. Bartolu be patient!'. Cosomeddu thinks like that: 'If I shoot at Antiogu first or at Bartolu, I'll kill one of them. It suits me better to shoot at Bartolu because he's a better shot and Antiogu might hit me with a 1/3 probability. It doesn't suit me to be the last one, because one of them could have in mind the good idea of shooting at me. If I'm the second one to shoot and Antiogu the first one, he'll hit me with a 1/3 probability; if, on the contrary, Bartolu is the first one there'll be a 2/3 probability. It suits me better to be the second one and Antiogu needs to shoot first, provided he understands he needs to shoot high.'
Bartolu thinks like that: 'It neither suits me to be the last one because if Antiogu misses Cosomeddu, he'll shoot at me; nor to be the second one because if Antiogu misses Cosomeddu this last one will shoot at me all the same. I must be the first one and shoot high, Cosomeddu must be the second one and so he shoots at Antiogu because Antiogu is the only one still armed. But Antiogu too will wish to be the first one for the same reason. So Antiogu and I have to agree and threaten Cosomeddu that we vote for him to be the last one and one of us can hit him. And it can be really worth for him to come to nothing this time. Damn him!'
And so the three duellers, each one for his own concern agree to defer to reckon with.";

continua