4. Sasha's conjecture is "All polygons with 6 equal sides are regular hexagons." Figure A is a counterexample because all 6 sides are equal but it is not a regular hexagon.
5. Bob's conjecture is "The difference between two numbers always lies between the two numbers."
The equation 6 - (-2) = 8 is a counterexample as 8 does not lie between 6 and -2, 8 is larger than both 6 as well as -2.
Didn't have time at first. So here is the other set:
3. The first pattern was constructed with multiples of 3. The next one with multiples of 3 x 3 = 9. The conjecture will deal with the next logical order: 3 x 3 x 3 = 27. Now we must answer: Is the sum of the digits of a multiple of 27, also a multiple of 27? (Hint, consider the number 54)
9. You've correctly identified the error to be in line 4. This is a calculation error.
13. The website linked below explains the frog puzzle very well and shows you how the number of moves can be modeled as (n + 1)^2 - 1 where n is the number of counters on one side. It follows then that for 5 counters on each side: (5 + 1)^2 - 1 = 36 - 1 = 35. So c is the correct answer.
And this is an interactive frog puzzle game you can mess around with the game itself: http://www.hellam.net/maths2000/frogs.html
23. I'm still working on it.
4. To find the counterexample look for the polygon that has six equal sides but is not a regular hexagon. So the answer is c.
5. The following is a counterexample because the difference between 6 - (-2) = 8 and 8 is not inbetween -2 and 6, but rather is greater than -2 and 6. So the answer is d
9. If x + y = z, then x + y - z = 0. That means you cannot divide both sides by (x + y - z) since that is division by zero; which is an error in calculation. So the answer is c.
16. Every even number can be written as the sum of two consecutive odd numbers. Proof:
Let n be an odd number. Then the next odd number is n + 2.
n + (n+2)
Add: 2n + 2
Factor out: 2 (n+1)
Any number multiplied by 2 is even.
17. The pattern can be written algebraically as:
6x + x = 7x which is true so the pattern will continue.
What abou number 3,9,13,23 note there is one 9 hasbeen answered