In
Mathcad, like in any other programming environment (or rather – in computer
environment), äà and
even in day-to-day life there are a lot of *defaults*.
A mother says to her son: «*Alex, please go for some butter!*» at that she
passes over in silence that he has to take some *money*, *bag* and buy *butter* in *the nearest shop* (and
so on and etc.).

The
main default, touching with vectors and matrix (arrays) in Mathcad, is in the
providential variable ORIGIN, which initial value equals zero.
It means that initial («upper») element of a vector (left column and upper line
of a matrix) has zero number. User has a right to break this default with help
of two methods. In the first place it possible by the command Options... from
menu Math to call the dialogue window Math Options with the laying Built-In
Variables, where in the field Array Origin the user can change zero to another
integer number (including negative numbers). In the second place it is possible
to write ORIGIN := N in Mathcad-document, where N – integer number, negative or positive one. But it is better not to
touch this variable at all trying to work with other ones. Generally speaking
it is necessary to have two variables of the type ORIGIN in order
to one of them keeps a number of a left column but the other – number of an
upper line and of the first element of a vector. But if we use only one
variable ORIGIN, then we’ll be able not to use this variable
at all (see the name of the tip), or rather to forget about it at all.

As
rule one changes the value of the variable ORIGIN from zero
to one. In that way he stresses that the numeration of the vector’s elements
begins usually from one but not from zero[1] in mathematics.

If
when you solve the problem and a vector is used then the numeration of the
elements begins from one and it is necessary not to use the variable ORIGIN, but you have to try to fill zero element up something.

Famous
puzzle «Hanoi’s towers» is solved in the document in the fig. 70. There are three sticks marked a, b and c. n mixed disk are
stringed on the stick a in imitation of child’s pyramid:
the biggest disk in the bottom of the stick but the smallest on – at the top.
It is necessary to move this pyramid from the stick a to the stick c using the following rules: 1) one disk is moved for one step; 2) the
stick b is used for intermediate warehousing of disks;
3) it is impossible to put a big disk on a small one.

The
solution of the problem for three disks is shown in the fig. 70: the function PD returns the line of textual constants that mark
the beginning of the game, the order of transposition and the end of the game.

Features of the program:

1.
In
general case the number of transpositions equals 2^{n}-1
– this number is noted in the variable p. The vector V that keeps the order of transpositions of the disks is created in
advance. We need that its initial element has had the first number and has kept
the direction of the first transposition. For this it is necessary to break the
default and to write ORIGIN := 1. But we shall not do it (*the main point of the tip*), but we’ll
fill zero element of the vector V up additional information about the
beginning of the game (“begin”). For «symmetry» we’ll fill the last (additional) element of the vector
up the message about the end of transpositions (“end”).

2.
The
problem about Hanoi’s towers is solved with help of *recursive* call of the function PD: the problem
with n disks is reduced to the problem with two
disks; the second of them is the disk that consists of n-1 disks. For
its turn the problem with n-1 disks is reduced to the problem
with two disks, the second of them is the disk that consists of n-2 and so on. This way it is possible to «go down» the problem with one
disk, which is solved simply – a-c. Hence there is recursion in the
program.

3.
The
solution of the problem about Hanoi’s towers in Mathcad is suggested by Êîíñòàíòèíîì Ìàìàåâûì from Áàðíàóëà.

[1] To begin the calculation of a
vector’s elements from one is European tradition, but from zero – American one.
The other «mathematical» difference of the Old and New World is the marking of
the imaginary unit: in our case it is i, but in USA – j.

«I have a comment about the TIP's footnote on
the last line. It's sometimes amusing to see that we don't always have
the same idea of what happens in other parts of the world. (1) In my
experience, vector and matrix indices beginning with 1 are much more common
than 0 in the U.S. although everyone understands that it's just an arbitrary
choice. and (2) At least in the U.S., most engineers use j for sqrt(-1), while
most physicists use i.»

Regards, Stan