[Archive!] Pure mathematics, physics, chemistry, etc.: brain-training problems not related to trade in any way - page 312

[Vladimir Gomonov]

Mathemat >>:
Richie, я могу привести пример на листе бумаги пять на пять, который не удовлетворяет требованиям задачи. Наверно, можно и для любого конечного, но тут помозговать надо.

Let's at least go 7x7 :)

[Sceptic Philozoff]

Richie, where are you? Will you draw a seven-by-seven square so that any number inside exceeds the numbers in at least five adjacent squares? We could pair up with Wasik.
Oh, I think I got 5 by 5 wrong: I don't have that example yet.

[Sceptic Philozoff]

Solution to problem 343:

There is nothing instructive about it. Hence, by the way, it follows that a 5 by 5 counterexample is unlikely to exist :) Not even 4 by 4 will work... And what does an endless sheet of paper have to do with it? A deliberately redundant datum introduced on purpose?

Next (I hope the class is clear?):

Apparently, "one of the heaviest" should be understood unambiguously: if there are several heaviest and the weight of each is the same, we choose any one.

[Alexey Subbotin]

Mathemat >>:
Решение задачи 343:

Ничего поучительного нет. Отсюда, кстати, следует, что контрпример 5 на 5 вряд ли существует :) Даже 4 на 4 не выйдет... И при чем тут бесконечный лист бумаги? Заведомо лишнее данное, внесенное специально?

Yeah, not good. It's a shame about the time wasted :(

[richie]

Sorry, very busy today and tomorrow.
Mathemat, for any number inside to exceed the numbers in at least 5 neighbouring cells - couldn't imagine that.
-
Here's some curious literature for "corrupting" kids:
//vse-knigi.su/auth/4652
//letitbit.net/download/8997.8966b0876903/Oster-ZadachnikPoMatematike.rar.html
-
Comrades moderators, fix the links at last.

[Sceptic Philozoff]

That's all right, Richie, we'll try to manage without you. Just let us know when you're free.
There's another one with weights in it:

P.S. It's probably assumed that weights can only be placed on one cup of the scale. The other has a weight on it.
One solution is obvious: 1, 2, 4, 8, 16, 32.

[richie]

Here is a funny program for finding options:
Suppose we have four weights, for example:
-
Dim MassaGiri1 As Long
Dim MassaGiri2 As Long
Dim MassaGiri3 As Long
Dim MassaGiri4 As Long
Private Sub Command1_Click()
For MassaGiri1 = 1 To 5
For MassaGiri2 = 1 To 5
For MassaGiri3 = 1 To 5
For MassaGiri4 = 1 To 5
If MassaGiri1 + MassaGiri2 + MassaGiri3 + MassaGiri4 = 2 * 3 Then
Print MassaGiri1, MassaGiri2, MassaGiri3, MassaGiri4
End If
Next MassaGiri4
Next MassaGiri3
Next MassaGiri2
Next MassaGiri1
End Sub
-
Run:

-
Choose any option, let's say 1:
So we have 3 weights of 1 kg and one of 3 kg .
-
1. Cup 1 =0 kg, cup 2 =0 kg;
2. On cup 1 we put 3 kg weight;
3. On cup 2 we put 1 kg weight;
4. On cup 2 we put 1 kg weight;
5. On cup 2 we put 1 kg weight;
-
The result is 3 kg on each cup . One option is proven.
-
Of course I understand that a formula is required, but there is no time right now.
Obviously it's just a progression where each successive term is equal to the sum of the previous ones:
1, 2, 3, 6, 12 etc.

[richie]

Here's the program for the 2nd task, sorry for the bad code, just no time:
-
Dim MassaGiri1 As Long
Dim MassaGiri2 As Long
Dim MassaGiri3 As Long
Dim MassaGiri4 As Long
Dim MassaGiri5 As Long
Dim MassaGiri6 As Long
Dim FileNo As Integer
Dim FileName As String
Dim Stroka As String
-
Private Sub Command1_Click()
FileName = "c:{Richi.txt"
FileNo = FreeFile
Open FileName For Output As FileNo
For MassaGiri1 = 1 To 32
For MassaGiri2 = 1 To 32
For MassaGiri3 = 1 To 32
For MassaGiri4 = 1 To 32
For MassaGiri5 = 1 To 32
For MassaGiri6 = 1 To 32
If MassaGiri1 + MassaGiri2 + MassaGiri3 + MassaGiri4 + MassaGiri5 + MassaGiri6 = 63 Then
If MassaGiri1 < MassaGiri2 Then
If MassaGiri2 < MassaGiri3 Then
If MassaGiri3 < Mass aGiri4MassaGiri4 Then
If MassaGiri4 < MassaGiri5 Then
If MassaGiri5 < MassaGiri6 Then
Stroka = Str(MassaGiri1) & Str(MassaGiri2) & Str(MassaGiri3) & Str(MassaGiri4) & Str(MassaGiri5) & Str(MassaGiri6)
Print #FileNo, Stroka
End If
End If
End If
End If
End If
End If
Next MassaGiri6
Next MassaGiri5
Next MassaGiri4
Next MassaGiri3
Next MassaGiri2
Next MassaGiri1
Close FileNo
End Sub
-
Prints a file of consecutive numbers, the sum of which is 63. The number of choices is large, the program hangs for a while.
Although, it's not quite clear what "consecutive" means.
Here's the beginning of the file (spaces replaced with hyphens):
-
1-2-3-4-21-32
1-2-3-4-22-31
1-2-3-4-23-30
1-2-3-4-24-29
1-2-3-4-25-28
1-2-3-4-26-27
1-2-3-5-20-32
1-2-3-5-21-31
1-2-3-5-22-30
1-2-3-5-23-29
1-2-3-5-24-28
1-2-3-5-25-27
1-2-3-6-19-32
1-2-3-6-20-31
1-2-3-6-21-30
1-2-3-6-22-29
1-2-3-6-23-28
1-2-3-6-24-27
1-2-3-6-25-26
1-2-3-7-18-32
1-2-3-7-19-31
1-2-3-7-20-30
1-2-3-7-21-29
1-2-3-7-22-28
1-2-3-7-23-27
1-2-3-7-24-26
1-2-3-8-17-32
1-2-3-8-18-31
1-2-3-8-19-30
1-2-3-8-20-29
1-2-3-8-21-28
1-2-3-8-22-27
1-2-3-8-23-26
1-2-3-8-24-25
1-2-3-9-16-32
1-2-3-9-17-31
1-2-3-9-18-30
1-2-3-9-19-29
1-2-3-9-20-28
1-2-3-9-21-27
1-2-3-9-22-26
1-2-3-9-23-25
1-2-3-10-15-32
1-2-3-10-16-31
1-2-3-10-17-30
1-2-3-10-18-29
1-2-3-10-19-28
1-2-3-10-20-27
......

See you tomorrow.

[richie]

Here are 4086 combinations (you have to save the file first, so it's all on one line):

[Sceptic Philozoff]

Richie, why don't you take care of yourself, you work so hard, by God, you write programs all the time in between jobs... It's a pain in the ass, isn't it? You can't do that. In the meantime, we'll take a break from the frenetic pace you're setting.