Weekend Puzzles...27_07_24

[imhotep]

Here are a few puzzles for the weekend to think about. Some are quite easy.

1. The Ducks. - There are two ducks in front of two ducks, two ducks behind two ducks and two ducks in between. What is the minimum number of ducks?
   
   
2. What four-digit number reverses itself when multiplied by 4? As in, what are the digits a, b, c and d such that the number abcd x 4 = dcba?
   The letters a, b, c and d all stand for different digits.
   
   
3. Shown below is a Trapezium with two parallel horizontal sides. Draw a vertical line in order to divide the shape into two parts of equal area?
   
   
   
   
   
4. Can you divide the trapezium equally by a line parallel to the bases?. If so, from where?
   
   
   

 

[EKGuest]

int x = 9999, a, b, c, d;

while (x-- > 0)
{
    a = x % 10;
    b = x % 100 / 10;
    c = x % 1000 / 100;
    d = x % 10000 / 1000;

    if (a != b && a != c && a != d && b != c && b != d && c != d)
    {
        int y = a * 1000 + b * 100 + c * 10 + d;
        if (y * 4 == x)
            Console.WriteLine(y + " * 4 = " + x);
    }
}

 

[imhotep]

Thanks.... You can solve this by a little bit of thinking too... Not brute force.
You know that 'a' has to be 1 or 2 , because anything above will result in a five digit answer when multiplied by 4.
Likewise you can build up to the answer by analysis.

 

[LZP1992]

Bump

 

[dilann]

1) 4
2) 2178

Spoiler

 

[EKGuest]

Thanks.... You can solve this by a little bit of thinking too... Not brute force.

Of course but as a slow thinker if there is a programmable solution I always go for it because it gets done much faster than if I do it manually.

 

[MrFrog]

bump

---

1. 4
2. 2178
- the number should be between 1000 and 2500, but 'a' cannot be 1 because 'd' cannot be 1, so 'a' is 2. 'd' should be 3 or 8, but as it should be even, 'd' is 8. 'b' cannot be 0 (number ending with 02 cannot be divisible by 4) , 2 (cannot repeat) or 4 (number ending with 42 cannot be divisible by 4). it should be 1 or 3. cannot be 3 if so it exceeds 8000 which doesn't agree with 'd', so 'b' is 1. Then solved for 'c'.

3. If the vertical line splits top line to a:b and bottom line to c:d, should they satisfy a + c = b + d to areas to be equal? Idk, may be there is a better way to express it.
------ Post added on Jul 27, 2024 at 7:47 PM

 

[imhotep]

bump

---

3. If the vertical line splits top line to a:b and bottom line to c:d, should they satisfy a + c = b + d to areas to be equal? Idk, may be there is a better way to express it.
------ Post added on Jul 27, 2024 at 7:47 PM

Dividing the trapezoid in two equal parts using a vertical line is not that difficult. However, dividing it using a horizontal (parallel to the bases) is not that easy.

 

[AnuradhaRa]

dividing trapezoid using a horizontal (parallel to the bases)
මේක කවුරුහරි හැදුවද
මාත් මේ ටිකේම ට්‍රයි කෙරුව
බැරුව ගියා
@imhotep

මේ ත්‍රිකෝණ කෑල්ල විතරක් වර්ගඵලය දෙකට බෙදලත් හරියන්නෙ නෑ නේද

 

[imhotep]

Nobody provided the answers for dividing the trapezoid.

Dividing the trapezoid by a vertical line is easy. Draw a line connecting the midpoints of the two non-parallel edges. Find the mid point of this line. Now draw a vertical line through this point.

Dividing it by a parallel line is a bit tricky. The length of this line is the RMS value of a & b - which is SQR{(a^2+b^2)/2}
This line can be geometrically constructed and it's said that the ancient Babylonians knew the trick.

PS: At least you are still working on the problem... Thanks!!!