Puzzles- contd

Quote:

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Originally Posted by rajraj

GP: :)

Let us move to triangles:

Two right angled triangles have a common hypotenuse (joined at the hypotenuse of same length). The other two sides of one triangle are 18 and 13 units. The other two sides of the second triangle are 20 and 7 units.
Obviously the units are not decimal (base ten numbers). What is the length of the hypotenuse and what is the base used?

All of you seem to be in IT sector. This should be stress relief for you! :)

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The base of the number system cannot be lesser than 9 (since there is a number with digit 8) at the same time it cannot be greater than 20, because if so all the numbers decimal equivalent would be the same (the biggest number is 20) and we already observe that the numbers don't meet the requirement in decimal base.

A tiny piece of Python code to iterate yields the answer to be base 12:

Code:

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def toDecimal( something,base ):
    index = 0
    number = 0
    for el in something[ : : -1 ]:
        number += int( el ) * ( base**index )
        index += 1
    return number

a1 = "18"
b1 = "13"

a2 = "20"
b2 = "7"

for i in range(9,20):
    print "Trying base %i" %i
    if toDecimal(a1,i)**2 + toDecimal(b1,i)**2 == toDecimal(a2,i)**2 + toDecimal(b2,i)**2:
        print "Success!!! The base is %i" %i
        break

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Quote:

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Trying base 9
Trying base 10
Trying base 11
Trying base 12
Success!!! The base is 12

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Quote:

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Originally Posted by rajraj

. --o--- o----o---o <---- Start here at (4,4)

.....o.....o.....o.....o

.....o.....o.....o.....o

.....o.....o.....o.....o

..........................<--- from (0,4) draw a diagonal line to (4,0)


You have drawn two lines so far connecting 7 dots!
Now use only vertical and horizontal lines to complete the solution! :) ( Remember crossing a line at a dot is permissible! )

This vaathiyaar is very patient but a little bit demanding ! :lol: Not a great artist ! :)

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Rajraj Sir,

Still I am unable to arrive at the solution :oops: Please post the complete co-ordinates.

Quote:

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Originally Posted by Benny Lava

The base of the number system cannot be lesser than 9 (since there is a number with digit 8) at the same time it cannot be greater than 20, because if so all the numbers decimal equivalent would be the same (the biggest number is 20) and we already observe that the numbers don't meet the requirement in decimal base.

A tiny piece of Python code to iterate yields the answer to be base 12:

Code:

---

def toDecimal( something,base ):
    index = 0
    number = 0
    for el in something[ : : -1 ]:
        number += int( el ) * ( base**index )
        index += 1
    return number

a1 = "18"
b1 = "13"

a2 = "20"
b2 = "7"

for i in range(9,20):
    print "Trying base %i" %i
    if toDecimal(a1,i)**2 + toDecimal(b1,i)**2 == toDecimal(a2,i)**2 + toDecimal(b2,i)**2:
        print "Success!!! The base is %i" %i
        break

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Quote:

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Trying base 9
Trying base 10
Trying base 11
Trying base 12
Success!!! The base is 12

---

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Correct! :) You can solve it using paper and pencil. Assume n to be the base ,express the numbers in decimal base and use Pythogorean theorem. You get the equation:

(n+8)^2 + (n+3)^2 = (2n)^2 + 7^2

or

n^2 -11n -12 =0

or

n=12

:)

You get an A- because you used a script ! :lol:

Quote:

---

Originally Posted by rajraj

. --o--- o----o---o <---- Start here at (4,4)

.....o.....o.....o.....o

.....o.....o.....o.....o

.....o.....o.....o.....o

..........................<--- from (0,4) draw a diagonal line to (4,0)


You have drawn two lines so far connecting 7 dots!
Now use only vertical and horizontal lines to complete the solution! :) ( Remember crossing a line at a dot is permissible! )

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line 1...................... (4,4) to (0,4) Connects the dots in top row

line 2...................... (0,4) to (4,0) Connects the dots below the main diagonal in the matrix of dots

line 3....................... (4,0) to (4,3) Connects the bottom three dots in the last column

line 4........................ (4,3) to (1,3) Connects the dots in the row below the top row

line 5......................... (1,3) to (1,1) Connects the bottom three dots in the first column

line 6......................... (1,1) to (3,1) Connects the first three dots in the bottom row

line 7..........................(3,1) to (3,2) Connects the bottom two dots in the third column

Benny Lava: I think you missed something in what I said or misunderstood something! :)

May be, I was not very clear ! :(

Quote:

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Originally Posted by rajraj

Correct! :) You can solve it using paper and pencil. Assume n to be the base ,express the numbers in decimal base and use Pythogorean theorem. You get the equation:

(n+8)^2 + (n+3)^2 = (2n)^2 + 7^2

or

n^2 -11n -12 =0

or

n=12

:)

You get an A- because you used a script ! :lol:

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Oh! Why didn't I think of that :oops:

Quote:

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Originally Posted by rajraj

line 1...................... (4,4) to (0,4) Connects the dots in top row

line 2...................... (0,4) to (4,0) Connects the dots below the main diagonal in the matrix of dots

line 3....................... (4,0) to (4,3) Connects the bottom three dots in the last column

line 4........................ (4,3) to (1,3) Connects the dots in the row below the top row

line 5......................... (1,3) to (1,1) Connects the bottom three dots in the first column

line 6......................... (1,1) to (3,1) Connects the first three dots in the bottom row

line 7..........................(3,1) to (3,2) Connects the bottom two dots in the third column

Benny Lava: I think you missed something in what I said or misunderstood something! :)

May be, I was not very clear ! :(

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Oh.. I was thinking that you were giving a solution with 6 lines. I was pretty convinced that it couldn't be done with any fewer than 7 lines.

No Rajraj sir, you were pretty clear.. it was just an assumption on my part that you were posting a solution for connecting with 6 lines.

OTOH, you said even in the original problem (9 dots version) the lines aren't supposed to cross. The solution that I have crosses each other. Could you please post the solution to that puzzle as well? :)

Benny Lava : Here is the solution for 3x3 dot matrix ! :)



.....o.....o.....o <- Start here at (3,3)

.....o.....o.....o

.....o.....o.....o

.................................



line 1...................... (3,3) to (0,3) Connects the dots in top row

line 2...................... (0,3) to (3,0) Connects the dots below the main diagonal in the matrix of dots

line 3....................... (3,0) to (3,3) Connects the three dots in the last column making a triangle

line 4........................ (3,3) to (0,0) Connects the dots in the other diagonal

You should have an arrow like figure! :)

Yes... this is what I have seen but in this solution line 2 and line 4 intersects at (1.5,1.5) hence my original confusion. :)

Quote:

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Originally Posted by Benny Lava

Yes... this is what I have seen but in this solution line 2 and line 4 intersects at (1.5,1.5) hence my original confusion. :)

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You are right! I forgot the restriction and gave you the solution for a simpler puzzle. With the added restriction the solution will resemble the letter Z , the letter S or the letter G requiring a minimum of five lines.

(That tells me that I should not do anything serious beyond midnight and three hours beyond my bed time! :lol: )

Ok :)