[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 339

[Ihor]

Mathemat >>:
Конечно, задача сильно усложняется и становится интереснее, если есть требование, чтобы решения были целыми. Хотя и здесь видна закономерность: положительные имеют вид 4к+1, 4к+2, 4к+3, 4к+4.

Yes with whole numbers the problem gets interesting.

Here are some more answers.

-1 3 5 24 46 -16560

-1 3 5 26 40 -15600

-1 3 5 28 36 -15120

-1 3 5 31 32 -14880

....

Can anyone find ALL the solutions ?

[Vladimir Gomonov]

ihor >>:

Может кто то может найти ВСЕ решения ?

I think this can only be possible if all solutions suddenly group into similar series.

Which in themselves are infinite, but can be described by "generators" as functions of natural k.

[Sceptic Philozoff]

I'm afraid there are a lot of parameterisations here, and not just single-parameter ones.

-1 3 5 24 46 -16560

-1 3 5 26 40 -15600

-1 3 5 28 36 -15120

-1 3 5 31 32 -14880

What's the parametrization here, I wonder?

2 avatara: I'm not going to contrast this thread with euroflood. Let it be exactly what it is - a more or less closed circle of interest.

[Sceptic Philozoff]

Two players play the following game: the first player writes the letters A or B in a row as he wishes (from left to right, one after the other; one letter per move), and the second player, after each move of the first player, swaps any two of the letters written out or changes nothing (this also counts as a move). After both players have made 1999 moves each, the game is over.
Can the second player play in such a way that any move by the first player results in a palindrome (i.e. a word that reads the same from left to right and from right to left)?

[oper]

Somebody had three apples. Somebody ate two apples...
To be continued. Such a greedy bastard...

[Tabletka]

The answer is yes. The permutations must be made with an odd number of letters and starting with the third, so that each permutation respects the palindrome

[Ihor]

qwerty1235813 >>:

Ответ - да. Перестановки нужно совершать при нечетном количестве букв и начиная с третей, так чтобы при каждой перестановке соблюдался палиндром

It doesn't work that way.

Let's say it was

ABABA or BAAAB.

add BB first

ABABA + BB

BAAAB + AA → no

BAAAB + BB

BBAAABB + AA → no

BABABAB +BB

BBBAAABB +AA → no

[Tabletka]

The thought process is as follows:
1)A+AB=ABA=ABA.
2)AV+AV=AVAVAV=AVAVAV
3) ABHABAB+AB=ABABHAB=ABHABEA (the third A is moved to the end of the word)
4)AVHABABAB+AB=ABHABAB=ABHABABAB (move the first letter to the middle).
etc.

[Tabletka]

But it doesn't say swap the nearby (those in close proximity) two.

[Ihor]

qwerty1235813 >>:

Но ведь не сказано что менять местами близлежайшие (те что в непосредственной близости) две.

in close proximity - not necessarily
but how to get a palindrome
BBAAABB + AA