🚀 go-pugleaf

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.   


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.


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?


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.