# Words with Puzzlers

My "friend" (who has absolutely too much time on his hands) made me a DIY Scrabble™ board. He's nice like that. He cut, sanded, painted and even routed out little cradles for the letter tiles.

Oh yeah, and he made letter tiles!

He left it for me as you see below, all filled in and scored on a scoring sheet. I noticed that the letter tiles had no values on them and examined the scoring sheet.

"Gadzooks and Whilakers! Bring me my head gear," I called, oozing masculinity, to my stunning, imaginary wife. "This must be some kind of puzzle for me to solve!"

I slowly, carefully unsheathed my Eagle Black Warrior pencil, sharpened it, and reviewed normal Scrabble™ scoring rules:

• Pink spots are double-word scores
• Red spots are triple-word scores
• Cyan and Blue respectively get double- and triple-letter scores.
• Simultaneous word multipliers get applied in succession
• Tiles that are already on the board which are used to make a new word (indicated by parentheses on the score sheet) get their unadulterated score (no letter- or word-multipliers are applied)
• Unlike normal Scrabble™, no bonus is applied for using 7 tiles in one turn.

Knowing my friend as intimately as I do, he probably wants me to discover the following:

1. What are the values for each of the letters?

2. What rule was used as the basis for assigning value to each letter?

3. Super-Secret Hidden BONUS: (2 words)

Scoring Sheet:

1.  STEIN             42            16. DI(M)             48
2.  (S)HRIMP          36            17. (P)ARA(NORMAL)    60
3.  (P)AT             16            18. (D)EL             18
4.  PLA(T)INUM       270            19. (L)EAVE           42
5.  (U)NITS           46            20. E(N)               9
6.  NITROU(S)         94            21. CO(P)PER          43
7.  (N)ORMAL          36            22. REA(C)TION        43
8.  (M)END            22            23. TAB(L)E           45
9.  TAUGH(T)          26            24. DIAGRA(M)        104
10. (T)AP             14            25. (A)CTIVE          30
11. EUREK(A)          72            26. ARS(E)NIC        123
12. (R)ARE            22            27. (E)LECTRON        71
13. APT (AE)          22            28. (T)ONGUE          86
14. TRE(E)            35            29. (TONGUES)STUN     67
15. MEE(T)            22            30. (G)ALAXY          86
31. ZINC(Y)           93


† The letters F,J,Q,W are not on the board, but their values must be determined for a complete solution.

• I don't see any red cells on the board. Where are they? (And what's with that singular white cell above the T in NITROUS?) – Deusovi Jun 20 '18 at 14:17
• Ignore the white cell I'll upload a fix. There are 8 "red" cells, the four corners and four center-side spots. – Chowzen Jun 20 '18 at 14:20
• Those are red? I thought they were pink! Where are the pink cells then? – Deusovi Jun 20 '18 at 14:21
• I assume the pink spots are the ones of the same color as the centre square? (along the two main diagonals and also 4 tiles from the centre) – Level 51 Jun 20 '18 at 14:25
• Is this a monitor issue? Adjacent to the 4 corners, for starters: Each corner is diagonally touching one of the (many) pink tiles. Also, the e in MEET, r in RARE. – Chowzen Jun 20 '18 at 14:26

[Extremely partial answer. Seems sufficient to be worth posting, though. Still contemplating.]

Letter values (part 1)

We have

a big pile of simultaneous equations to solve. I confess that I got my computer to do it. The results are as follows:
$\begin{array} \\ A:2 && B:4 && C:6 && D:10 && E:2 \\ F:\,? && G:3 && H:1 && I:3 && J:\,? \\ K:10 && L:3 && M:3 && N:7 && O:8 \\ P:6 && Q:\,? && R:9 && S:5 && T:4 \\ U:11 && V:5 && W:\,? && X:2z-3 && Y:15-z \end{array}$
where $z$ is the value of $Z$: so X,Y,Z could be 5,14,1; 7,13,2; 9,12,3; 11,10,4; ...; 31,1,14.

Where do they come from?

I don't know yet. I have dark suspicions that

chemistry is involved, and that perhaps the secret two words might be PERIODIC TABLE (look at all those elements on the board! see how H=1 and C=6, and try to overlook the values of N,O,P,S,K,I,V!) [EDITED to add:] D'oh, evidently I was right to be looking at H and C ... and I'm not sure what variety of brain-failure made me think N and O were problems. Well done to Reinier for seeing what I missed there; go upvote his answer.

but my concrete reasons for thinking so are mostly nonsense, and the details mostly fail to check out.

Presumably once their origin is made clear, the values of FJQW will be obvious.

Some further notes on hidden things on the board:

DIM/MEET/TREE MEND/DEL/LEAVE (Dmitri Mendeleev) -- perhaps these are the super-secret words.
Obvious elements: arsenic, copper, platinum, zinc(y), nitrogen (well, nitrous). Not quite so obvious: TONGUES/STUN.
Other chemistry-themed things: REACTION, ELECTRON.
TABLE.
There's RARE but not EARTH; maybe coincidence.

• Darn it! I solved by hand, but didn't finish quite as quickly. I can confirm that I got the same results, though. – GentlePurpleRain Jun 20 '18 at 16:17
• The secret words have been found. I'll call this the accepted answer, with kudos to @Reiner for the balance of the letter values and to@GentlePurpleRain for doing it by hand. – Chowzen Jun 20 '18 at 16:58

Partial answer, using the findings of Gareth McCaughan

the periodic table is used in finding the values of the letters. For each letter, the first element which symbol contains this letter is used. Then the digit sum of the atom number is taken, which is the value of the letter. (So for A, the first element is Na, with atom number 11, so the value is $1+1 = 2$.) We get the following results:

A Na 11 2
B Be 4  4
C C  6  6
D Pd 46 10
E He 2  2
F F  9  9
G Mg 12 3
H H  1  1
I Li 3  3
J ??
K K  19 10
L Li 3  3
M Mg 12 3
N N  7  7
O O  8  8
P P  6  6
Q ??
R Ar 18 9
S Si 14 5
T Ti 22 4
U Cu 29 11
V V  23 5
W W  74 11
X Xe 54 9
Y Y  39 12
Z Zn 30 3


These values completely agree with the values found by Gareth McCaughan. However, there is still a problem:

There are namely no element symbols containing J and Q (or I looked over them), so something else has to be done to find their values.

• Ah! I never thought of digit-summing. I might have, if there hadn't been numbers bigger than 9 (because in my head, any time you add up digits, you should be iterating that until you get a single digit -- standard technique for reducing modulo 9). Duh :-). – Gareth McCaughan Jun 20 '18 at 16:25
• Arguably the value for Q should be 1+2+4=7, but that doesn't help resolve J. [And it has the problem that at some time in the future Q might have to change to 1+3+4=8.] – Gareth McCaughan Jun 20 '18 at 16:28
• Maybe the remaining letters still follow the rule, and they'd therefore hold the lowest value? – hagfy Jun 20 '18 at 16:29
• In retrospect, maybe the Q and J values weren't as blatantly obvious as I originally thought. I simply added the "digit sum of the atom number" of the element with each letter. No element = no number = 0. "Unbiquadium" was not on my radar. :) – Chowzen Jun 20 '18 at 16:48
• Hm, OK. So @Chowzen, between Reinier and me, are we done? E.g., did I find the correct secret two words? Or is there more yet to do? – Gareth McCaughan Jun 20 '18 at 16:54