# What do these things have in common?

There are three sets of these puzzles you are required to answer!

Set 1:

Buck
Baby
Wisdom

Set 2:

Scavenger
Fox
Treasure

Set 3:

Year
Scrap

Remember; you need to get all three!

• You say, “There are three sets of these puzzles you are required to answer!”  I would quibble with that: there are three puzzles, not three sets of puzzles (there are, arguably, three sets of clues).  But I'm making the point that, lacking a link, these are three separate puzzles, and I don't see any good reason to post them together in one question. (Then again, if you posted them as three separate questions, they would probably get downvoted for being to easy.  I solved all three in about a minute, once I got around to reading this question.) – Peregrine Rook Jul 31 '16 at 2:39

Set 1

buck-tooth, baby-tooth, wisdom-tooth

Set 2

scavenger-hunt, fox-hunt, treasure-hunt

Set 3

• Should I be finding a meta-link? – Jonathan Allan Jul 29 '16 at 22:42
• what do you mean? – that2guy Jul 29 '16 at 22:43
• I mean should I be looking for a similar way to link the three solutions? – Jonathan Allan Jul 29 '16 at 22:44
• no, you got it. – that2guy Jul 29 '16 at 22:51
• I wish the original puzzle took longer to figure out, but I will occupy myself with finding a satisfying way to link "tooth hunt book" in the mean time... – tmpearce Jul 30 '16 at 1:31

Set 1:

TOOTH

Set 2:

HUNT

Set 3:

BOOK

We can combine these to form

this alphametic: \begin{align}\text{TOOTH}\\-\quad\text{HUNT}\\\hline\text{BOOK}\end{align} i.e., TOOTH − HUNT = BOOK
(because we obviously can’t have TOOTH + HUNT = BOOK)

Or, equivalently,

BOOK + HUNT = TOOTH

BOOK < 9999 and HUNT < 9999, so TOOTH < 19998.  Therefore, T ≤ 1.  If we make the standard assumption that numbers don’t have leading zeroes, then T = 1.  Similarly, B ≠ 0 and H ≠ 0.  Making the standard assumption of a unique mapping, H ≠ 1, and, if H = 2, then K = 1 (from the rightmost column0: K + T = H (with possible carry); i.e., K + 1 = H or K = H − 1), which also violates uniqueness.  So H ≥ 3.
______
0 I’ll refer to the rightmost column (the units column) as column 0 because it represents the 100 values.  Moving to the left, the columns are 1, 2, 3, and 4.

So, anyway, column 0 cannot carry (because H cannot be 0, and K + 1 cannot be larger than 10).  In column 1 we have O + N = T = 1 (with possible carry).  Neither O nor N can be 1, and so neither of them can be zero (i.e., O ≥ 2 and N ≥ 2), so this sum must carry (i.e., O + N = 11, so N = 11 − O).  So, in column 2, 1 (carry) + O + U = O (with carry), so U = 9 and column 2 carries.  And, since N ≠ 9 (because U = 9), N ≤ 8 and O ≥ 3.

Looking at column 3, 1 (carry) + B + H = 10 + O; i.e., B + H = 9 + O (or B = 9 + O − H).  B ≠ 9 (because U = 9), so B < 9, so H > O.  And O cannot be H − 1 (because that’s K), so O < H − 1; i.e., O < K.  Since O ≥ 3, we have K ≥ 4 and H ≥ 5.

I had to go to brute force here.

Suppose H = 5 and K = 4.  Since O < K, O must be 3 and N = 8.  B = 9 + O − H = 7, giving us a solution: \begin{align}13315\\-\quad5981\\\hline7334\end{align}

Suppose H = 6 and K = 5.  We can have O = 3 and N = 8, or O = 4 and N = 7.  O = 3 gives us B = 9 + 3 − 6 = 6, which equals H and so is a contradiction.  O = 4 gives us B = 9 + 4 − 6 = 7, which equals N and so is a contradiction.

Suppose H = 7 and K = 6.  We can have (O,N) = (3,8), (4,7) or (5,6) — but the second and third choices give us N equal to either H or K.  (3,8) gives us B = 9 + 3 − 7 = 5, which gives us the solution \begin{align}13317\\-\quad7981\\\hline5336\end{align}

Suppose H = 8 and K = 7.  We can have (O,N) = (3,8), (4,7), (5,6) or (6,5) — but the first and second choices give us N equal to either H or K.  (5,6) gives us B = 9 + 5 − 8 = 6, which equals N and so is a contradiction.  O = 6 gives us B = 7, which equals K and so is a contradiction.

(H cannot be 9 because U = 9.)

P.S. There is a trivial alternative to the first puzzle:

TEETH.  It turns out that TEETH - HUNT = BOOK has eight solutions.

OK, yeah; I suppose I could say HUNTS or BOOKS, too; I didn’t even look at those.