by Guy Jacobson
Problem:

The “math” problems here do not involve numerical addition and subtraction, so don’t break out your calculators. What the “familiar character” actually says to you is “elementary”; this and the range of numbers (1–118) suggests chemical elements. Solvers need to treat these problems as rebuses involving the chemical symbols for the elements with the given atomic numbers:

 112 + 107 − 41 + 33 = CN + BH − NB + AS CHAS 16 + 67 + 16 = S + HO + S SHOS 14 + 116 = SI + LV SILV 89 + 105 − 48 + 4 = AC + DB − CD + BE ABBE 97 − 19 + 68 + 39 = BK − K + ER + Y BERY 27 + 15 + 15 = CO + P + P COPP 66 + 49 = DY + IN DYIN 99 + 96 − 21 + 78 = ES + CM − SC + PT EMPT 63 − 92 + 7 + 31 + 38 − 33 = EU − U + N + GA + SR − AS ENGR 53 + 116 − 23 + 71 = I + LV − V + LU ILLU 25 + 94 − 93 + 51 + 111 − 35 = MN + PU − NP + SB + RG − BR MUSG 102 + 37 − 5 + 74 = NO + RB − B + W NORW 75 + 22 = RE + TI RETI 34 + 20 + 42 − 95 = SE + CA + MO − AM SECO 9 + 53 + 23 + 2 − 1 = F + I + V + HE − H FIVE 57 + 38 + 85 − 88 = LA + SR + AT − RA LAST 1 + 8 + 29 − 6 + 7 = H + O + CU − C + N HOUN 90 + 118 + 18 − 31 = TH + OG + AR − GA THOR 45 + 32 − 80 + 110 + 113 − 50 = RH + GE − HG + DS + NH − SN REDH 23 + 13 + 71 − 92 = V + AL + LU − U VALL

Note that the rebuses all lead to somewhat word-like four-letter sequences, but what are they? Solvers need to use their powers of deduction to identify the “familiar character” who says “elementary”. Of course, he is none other than Sherlock Holmes, and these four-letter sequences are all standard abbreviations for short stories and novels in the Sherlockian canon.

The next step is to expand these abbreviations into the full titles of the stories (the rather wide blank spaces suggest that something long belongs there), and use the two numbers in parentheses after the blanks as indices into those titles, selecting two (or, in one case, one) letters, forming another set of element symbols:

Abbr. Full Title Indices Symbol
CHAS The Adventure of Charles Augustus Milverton (38, 16) NH
SHOS The Adventure of Shoscombe Old Place (22, 16) BH
SILV The Adventure of Silver Blaze (24, 20) ZR
ABBE The Adventure of the Abbey Grange (18, 23) AG
BERY The Adventure of the Beryl Coronet (27, 18) NB
COPP The Adventure of the Copper Beeches (18, 19) CO
DYIN The Adventure of the Dying Detective (25, 27) TC
EMPT The Adventure of the Empty House (19, 21) MT
ENGR The Adventure of the Engineer’s Thumb (25, 28) RH
ILLU The Adventure of the Illustrious Client (29, 27) CU
MUSG The Adventure of the Musgrave Ritual (22, 25) RE
NORW The Adventure of the Norwood Builder (21) W
RETI The Adventure of the Retired Colourman (31, 33) MN
SECO The Adventure of the Second Stain (20, 22) CN
FIVE The Five Orange Pips (5, 9) IR
LAST His Last Bow (7, 2) TI
HOUN The Hound of the Baskervilles (4, 10) HF
THOR The Problem of Thor Bridge (5, 21) RG
REDH The Red-Headed League (9, 17) AU
VALL The Valley of Fear (15, 12) RF

Now, what to do with this new set of element symbols? Solvers need to mark these elements on a standard periodic table:

We can see the puzzle answer, NIL, spelled out in the marked cells.