🚀 go-pugleaf

Joseph <cschen@xnet.com> wrote:

>Can anyone solve the following puzzles ?
>1  Replace the first digit of one of the integers 5,471,602 and
>   6,650,864 and the last digit of the other integer with the 
>   same digit to make each integer divisible by 6.   

Well, all multiples of 6 have the following properties: The digits all
add to a multiple of 3, and the numbers are even.  Add the interior
digits together, and the first sum is 18, and the second is 25.  Now,
assuming that you change the first digit of the first number the sums
before the change and including the remaining digit would be 20 and
31, which disallows that choice because no single integer can be added
to both an odd and an even number and make both sums even.  This means
that the you must change the last digit of the first number and the
first digit of the second, giving incomplete sums of 23 and 29, which
are both even.  Looking at the numbers, 4 will work because the sums
will then be 27 and 33, both multiples of 4.

>2  In the sequence of numbers 1,3,2,.....each term after the 
>   first two is defined to be equal to the term preceding it
>   minus the term preceding that. Find the sum of the first 1000
>   terms of the sequence.

If you work out the first few terms, you'll note that it is a
repeating pattern, and it looks like this:
1,3,2,-1,-3,-2,1,3,2....... and so, each set of 6 cancels itself.
Divide 1000 by 6 and you get 166 2/3, so there are 166 full iterations
which add to 0, and 2/3 of an iteration which would be 1,3,2,-1,
giving a sum of 5.

>3  A number of objects are stacked in a triangular pyramid
>   (1 in the top layer, 3 in the second layer, 6 in the third layer,
>   10 in the fourth layer, etc.).
>       a. How many objects are in the nth layer from the top?
>       b. How many objects are in the top n layers?

Each layer would have 1+2+3+4+....(n-2)+(n-1)+n objects in it, which
means that the rate of change at n levels is n, so the formula for the
change per level is f(x)=n, so the antiderivative would be 1/2n^2

>4   A,B,C,D,E,F,  and G are having an argument about which day of the 
>    week it is. They speak as follows:
>	A:  The day after tomorrow is Wednesday.
>	B:  No, it is Wednesday today.
>	C:  You are both wrong; it is wednesday tomorrow.
>	D:  Nonsense.  Today is neither Monday, Tuesday, nor Wednesday.
>	E:  I'm quite sure yesterday was Thursday.
>	F:  No, tomorrow is Thursday.
>	G:  All I know is that yesterday was not Saturday.
>   If only one of the remarks is true, what day of the week is it?



>5   You have 8 metal spheres, all of which look exactly the same. 
>    Seven of these weigh the same and one is either heavier or lighter 
>    than the others.  You have a balance scale.  Tell how you could 
>    find the odd sphere in at most three weighings and explain how 
>    you can decide whether the odd sphere is heavier or lighter than 
>    the others.


>6  Are there two different prime numbers p and q such that p+q 
>   divides pq evently: if so, give an example.  
>   If not, prove why not.


>7  Two concentric circles are given.  A tangent to the inner circle 
>   forms a chord twelve inches long in the larger circle. 
>   Find the area of the ring between the circles.

 - Executioner/[Independent]
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