Random Puzzles 

Workings

Hint
This is a very common sequence.

Answer
X.

The last letters of the words one, two, three, four, five, six.

Workings

Hint
Try removing the spaces from the sequence.

Answer
1301311 or 1210121.

Reasoning
Removing spaces from the original sequence gives:

   1 23 124 1251 26127 128129

   123124125126127128129

Now adding spaces back in, in the correct place:

   123 124 125 126 127 128 129

The next few terms could be:

   130 131 132
   or
   1210 1211 1212

Add these to the end:

   123 124 125 126 127 128 129 130 131 132
   or
   123 124 125 126 127 128 129 1210 1211 1212

Remove the spaces again:

   123124125126127128129130131132
   or
   123124125126127128129121012111212

Now break the sequence into chunks of increasing size:

   1 23 124 1251 26127 128129 1301311
   or
   1 23 124 1251 26127 128129 1210121

So the missing term is 1301311 or 1210121.

Workings

Hint
Be careful with your thinking.

Answer
Exactly the same time!

Workings

Hint
Remember that Time = Distance ÷ Speed, and try setting the track length to be a random length.

Answer
119mph.

Reasoning
It doesn't matter how long a lap is, so let's set the track length to be a random number, say 5 miles.

Using:
   Time = Distance ÷ Speed

The first two laps took:
   ¹⁰/₁₀₀ hours

The second two laps took:
   ¹⁰/₁₀₂ hours

The next two laps took:
   ¹⁰/₁₄₀ hours

The final two laps took:
   ¹⁰/₁₅₀ hours

Therefore, the total time for the 8 laps was:
   ¹⁰/₁₀₀ + ¹⁰/₁₀₂ + ¹⁰/₁₄₀ + ¹⁰/₁₅₀ hours
   = ¹/₁₀ + ⁵/₅₁ + ¹/₁₄ + ¹/₁₅ hours

Looking at the denominators:
   10 = 2 x 5
   51 = 3 x 17
   14 = 2 x 7
   15 = 3 x 5

The lowest common multiple (LCM) of these numbers is 2 x 3 x 5 x 7 x 17 = 3570.

   = ³⁵⁷/₃₅₇₀ + ³⁵⁰/₃₅₇₀ + ²⁵⁵/₃₅₇₀ + ²³⁸/₃₅₇₀ hours
   = ¹²⁰⁰/₃₅₇₀ hours
   = ¹²⁰/₃₅₇ hours
   = ⁴⁰/₁₁₉ hours

The 8 laps were 40 miles, and using Speed = Distance ÷ Time:
   average speed = 40 ÷ (⁴⁰/₁₁₉)
   average speed = 40 x (¹¹⁹/₄₀)
                 = 119 mph