You find a forgotten list on an ancient piece of paper while cleaning your professor's office. It has the following written down:

Exactly 1 statement on this list is false.

Exactly 2 statements on this list are false.

Exactly 3 statements on this list are false.

Exactly 4 statements on this list are false.

How many statements on the list are in fact false?

Find the value of

\[ \frac { { \left( { 3 }^{ 2017 } \right) }^{ 2 }-{ \left( { 3 }^{ 2015 } \right) }^{ 2 } }{ { \left( { 3 }^{ 2018 } \right) }^{ 2 }-{ \left( { 3 }^{ 2016 } \right) }^{ 2 } }.\]

A ball approaches a batsman horizontally at speed \(v_{\text{ball}}\), and the batsman swings the bat, hitting the ball back along the same path. If the speed of the bat is \(v_{\text{bat}}\) when it hits the ball, then what is the recoil speed of the ball?

\(\)

**Details and Assumptions:**

- The collision is perfectly elastic.
- For all practical purposes, the speed of the bat is constant throughout the collision.

**True or False?**

For any integer \(n>3\), the last digit of \[\large 2^{1\times 2\times 3\times \cdots \times n}\] is always 6.

Can we place a distinct integer from 1 to 8 into each black circle such that the sums along the 7 colored paths are the same?

For example, we have filled up the circles, but not all of the colored paths have the same sum.

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