Random Puzzles 

Workings

Hint
If we look at all of the clues, each person appears exactly twice.

Answer
Alex had been teaching for 21 years.
Billie had been teaching for 15 years.
Charlie had been teaching for 7 years.

Reasoning
If we look at all of the clues, each person appears exactly twice.

So the total of the three clues gives us two lots of Alex + Billie + Charlie = 36 + 22 + 28 = 86.

So, Alex + Billie + Charlie = 43 years.

In each clue we are given two of the people, so we can use this to find the missing person:

   Charlie and Billie had been teaching for a total of 22 years, which means that Alex must be: 43 − 22 = 21 years.

   Charlie and Alex had been teaching for a total of 28 years, which means that Billie must be: 43 − 28 = 15 years.

   Alex and Billie had been teaching for a total of 36 years, which means that Charlie must be: 43 − 36 = 7 years.

Reasoning With Algebra
Let Alex = A, Billie = B and Charlie = C, then:

[1]   A + B = 36
[2]   C + B = 22
[3]   C + A = 28

If we use [3] − [2] we have:

[4]   A − B = 6

If we use [1] + [4] we have:

   2A = 42
    A = 21

By [1] B = 15. By [3] C = 7.

Workings

Hint
Rewrite the clues so that those who finished ahead are written first.

Answer
Crystal Ball        (CB)
 Justin Time        (JT)
  Inga Hootz        (IH)
   Leigh Vamessage  (LV)
    May Eye         (ME)
     Mike Stup      (MS)
      Laura DeLand  (LD)
       Joe King     (JK)
        Holly Day   (HD)
         Jack Pott  (JP)

Reasoning
If we number the clues, write each name using two letters, and rewrite the clues so that those who finished ahead are written first:

CB beat ME and MS LV beat ME and LD HD, MS, and JT, all beat JP MS, ME, and CB, all beat LD JK beat HD and JP LD and IH both beat JK CB and IH both beat LV JT beat IH CB beat JT JT beat HD and JP ME beat MS and HD LV and IH both beat HD
By (9): CB JT
By (8): CB JT IH
By (7): CB JT IH LV
By (2): CB JT IH LV ME
By (11): CB JT IH LV ME MS
By (4): CB JT IH LV ME MS LD
By (6): CB JT IH LV ME MS LD JK
By (5): CB JT IH LV ME MS LD JK HD
By (3): CB JT IH LV ME MS LD JK HD JP

Workings

Hint
The 8-letter word begins with the letter G.

Answer
Generous.
lend + ge = le + ge + nd = legend

liar + ne = li + ne + ar = linear

pale + ro = pa + ro + le = parole

coin + us = co + us + in = cousin

Workings

Hint
Try starting in one corner.

Answer
There is no possible route.

Leonhard Euler (1707 to 1783), a Swiss polymath, was the first to prove that this puzzle had no solution.