Brain Teasers and Puzzles 

Workings

Hint
Could Alex Ambleton have won?

Answer
Taylor Tripton won with Midday Night.

Reasoning
Before we begin the reasoning, let's look at the hint: 'Could Alex Ambleton have won?'. There are three horses in the puzzle, and Alex Ambleton is stated to have won with all three of them. We know that none of the statements were fully true, so Alex couldn't have won otherwise one of the statements would have been fully true.

By grouping the clues together, we have:

1. Frankie Flanton won with Morning Sunset.
2. Frankie Flanton won with Evening Sunrise.
3. Frankie Flanton won with Evening Sunrise.

4. Sam Spinton won with Midday Night.
5. Sam Spinton won with Midday Night.
6. Sam Spinton won with Midday Night.
7. Sam Spinton won with Morning Sunset.

8. Taylor Tripton won with Morning Sunset.
9. Taylor Tripton won with Evening Sunrise.

10. Alex Ambleton won with Midday Night.
11. Alex Ambleton won with Morning Sunset.
12. Alex Ambleton won with Evening Sunrise.

We know that none of these statements were fully true, so, we know that (for example) it wasn't Frankie Flanton with Morning Sunset.

We can create a grid of all the possibilities and place an X where we know that it can't have been that combination:

We now have an X for all but 3 possibilities, and we know that exactly 6 of the clues must be half true.

So…

…if it was Frankie Flanton with Midday Night then exactly 7 clues would be half true (clues 1, 2, 3, 4, 5, 6, 10).

…if it was Sam Spinton with Evening Sunrise then exactly 8 clues would be half true (clues 4, 5, 6, 7, 2, 3, 9, 12).

…if it was Taylor Tripton with Midday Night then exactly 6 clues would be half true (clues 8, 9, 4, 5, 6, 10) – so this must be the correct answer.

Workings

Hint
The first letters of the words are: D, D, F, I, L, L, P, S, S, S.

Answers
dif + fer + ent = different
dig + est + ive = digestive
far + mho + use = farmhouse
imp + ort + ant = important
lib + rar + ian = librarian
lig + htn + ing = lightning
por + tfo + lio = portfolio
sca + rec + row = scarecrow
sin + cer + ely = sincerely
som + eth + ing = something

Workings

Hint
The answer is in front of you.

Answer
Mississippi.

The diagram is actually the word "mississippi" on its side.

Workings

Hint
As we don't know how of the people were library members, you can try a random number and see what happens.

Answer
4,200 books.

Reasoning
It doesn't matter how many library members there were, as ¹/₂ of them accepted the books, and each accepted 6 books, which is 3 books per member on average.

Similarly, it doesn't matter how many non-members there were as ³/₄ of them accepted the books, and each accepted 4 books, which is again 3 books per non-member on average.

Double-Checking
Both the members and non-members accepted, on average, 3 books each.

And 1400 x 3 = 4200.