Thanks to all of you who were interested in the very first S&SMC, posted

to this forum a week or so ago. Its thread had a total of 17 messages

posted (including 3 by myself), so I take this to mean you like

the 'section'. Special thanks go to Mr. Andres C. Rodrigues for his enthusiastic

support and highly encouraging words. To honour your request, here's a new challenge

for all of you and your favourite HP handhelds.

Last week's challenge (the one about the tangents), was mostly theoretical

in nature, with just a little programming being useful to empirically try and

discover the underlying relationship (a+b+c=180). Once postulated, said

relationship could be demonstrated using symbolic math, perhaps with the

help of some able HP (such as an Hp48 or HP49), though unfortunately nobody

commented on whether those advanced models could cope with it and how.

All in all, it was a rather theoretical (if interesting) challenge, so in order

to compensate somewhat, this week's challenge, #2, is much more empirical and

will have you and your HPs working hard in order to succeed. It goes

like this:

- 1) Take your favourite HP calc which can do matrices, either using fast

built-in capabilities or else suitable programs for the task at hand.

Specifically, it must be capable of storing a 3x3 or 4x4 matrix and

compute its determinant.

(Models with that capability built-in in fast microcode include: HP-15C,

HP-41C + Advantage ROM, HP-71 + Math ROM, HP-42S, and HP-48/49 models,

among several others. Most models can also be easily programmed to

compute determinants up to 3x3 at least, so you've got ample choice)

- 2) Now, you must define a 3x3 matrix called A, and fill it up with the integers from 1 to 9,

in any order you like as long as none of them are repeated. For instance,

you can have your matrix like this:| 6 1 8 |

A = | 7 5 3 |

| 2 9 4 |

and its determinant is det(A) = 360 - 3) The challenge is: find the arrangements of the integers from 1 to 9 which:
- make det(A) the minimum possible non-zero value.

- make det(A) the maximum possible value.

We are only interested on absolute values of the determinant, regardless of the sign.

The solutions are not unique, of course. - make det(A) the minimum possible non-zero value.
- 4) As you can see, there are fact(9) = 362880 possible arrangements

of the integers 1,2,...,9 without repetitions, so you'll have to

make good use of your ingenuity and your HP's capabilities if you

intend to avoid very long computation times. - 5) If you succeed in the 3x3 case, try your might with the 4x4 case,

i.e: find the arrangements of the integers from 1 to 16 which

make the determinant of the 4x4 matrix maximum or minimum (but non-zero).

The number of possible arrangements is now fact(16), an impossibly large

number to use brute force. Some finesse is indeed required.

For instance, can you find an arrangement which makes det(A) = 1 ? Or = 4000 ?

or = 4444 ? Or > 40000 ? Which is more, can you

theoretically demonstrate what are the maximum/minimum

values of the determinant for an NxN matrix ?

Give it a chance, refrain from the temptation to use a pc, and

let us all see how you and your HP cope with this challenge !