The Square Root of 576 by Leo
This week Leo gave us a puzzle based on a fascinating discovery.

Extra material in answers turned out to be words or phrases associated with numbers. Intersecting answers had different material leading to the same number. The corresponding clue answers were one letter too short for their allotted positions, leaving some cells unfilled. When the numbers associated with the extra material were written in the empty cells, a 3 by 3 magic square was formed, with rows of 25, 8, 12; 2, 15, 28; and 18, 22, 5.

The message spelled out by the extra letters in wordplay was LETTER COUNT GIVES A SECOND MAGIC SQUARE. Counting the number of letters in each element of the first magic square gave the following: TWENTY FIVE (10), EIGHT (5), TWELVE (6); TWO (3), FIFTEEN (7), TWENTY EIGHT (11); EIGHTEEN (8), TWENTY TWO (9), FIVE (4). Remarkably, the letter counts also formed a 3 by 3 magic square, with the sum of each row, column and diagonal equal to 21. When TWENTY ONE was written in the cells which were originally empty, in grid order, new words were formed, and this was the grid to be submitted. Gregson noted that TWENTY TWO, THIRTY ONE and THIRTY TWO would also lead to real words, but these were red herrings carefully planted by Leo.

The significance of the title could be explained in two ways. The square root of 576 is 24, and there are 24 letters in FIVE HUNDRED AND SEVENTY SIX. Furthermore, 24 was the difference between the row totals of the two magic squares, ie 45 – 21.

Gregson was left wondering if any other magic squares exhibit this type of duality.

Extra material in clues, and extra letters in wordplay, are indicated by [].

1 Many tracks are found here but no cutting edge (4)
E[L]PEE Two meanings (one with extra letter)
4 Job has support immediately (9)
POSTHASTE[E] POST (Job) + HAS + TEE (support)
11 Test a tense fortification (4)
MO[T]AT MOT (Test) + A + T (tense)
13 Singer murdered chart tune? (8)
14 Province of real-ale man (not boozy Alan?) (5)
REALM[E] REAL-ALE MAN less anagram of ALAN
15 Carry rich cake (4)
TO[R]TE Two meanings (one with extra letter)
16 [Line on pitch (TWENTY-FIVE)] happening to enclose box (5)
CASE Two meanings
17 [Pieces of (EIGHT)] soaked bread, very soft (4)
SOP SO (very) + P (soft)
18 Wandering about found [(TWELVE) Roman tables] in Mogadishu (4)
GAD Hidden in MoGADishu
19 Slack about miscellaneous seconds (6)
REMIS[C]S RE (about) + MISC (miscellaneous) + S (seconds)
24 Like loose woman on the game, or one associated with love? (7)
ASTARTE[O] AS (like) + TART (loose woman) + E O (game)
25 Ceremonial robe and British hat - clothing indeed! (6)
[U]KHALAT UK (British) + HAT containing LA (indeed)
26 The Eastern one[(TWO)'s company] for one mentioned often with sympathy (4)
TEA T’ (The) + E (Eastern) + A (one)
28 [RU team (FIFTEEN)] regularly maul pair of wings (5)
31 Tree seen bending back in [perfect (TWENTY-EIGHT)] deluge (4)
ULE Hidden, reversed, in dELUge
28 being a perfect number, ie one whose divisors sum to the number itself (1,2,4,7,14)
32 Disguised meaning (6)
ENIGMA[N] Anagram of MEANING, & lit
33 A tax on grain betrays greed (7)
AVA[T]RICE A + VAT (tax) + RICE (grain)
38 Terrible gulag with no European water (6)
[G]LUGANO Anagram of GULAG, + NO
39 What preceded IBM in 2001 – [majority (EIGHTEEN)] had late date unfortunately missing (4)
HAL, the computer in the film 2001 – A Space Odyssey, was named by taking the letters preceding IBM
41 [Two little ducks (TWENTY-TWO)] paddle furiously, not half under control! (4)
LED Anagram of DLE (second half of PADDLE)
“Two little ducks” being a bingo call for the number 22
43 Save game from dryness, [famous (FIVE)] reversal of what's dull (5)
BARD DRAB (dull), reversed
45 Big biography is written about African (4)
IGB[I]O Hidden in bIG BIOgraphy
46 Audio-visual artist touches soul in Eastern conveyance (5)
A[V]RABA AV (Audio-visual) + RA (artist) + BA (soul)
47 One to look back and have look out (8, two words)
KEEP[E]CAVE ACE (One) + PEEK (to look), all reversed, + ‘VE (have)
48 See fast bird use me on the rebound (4)
EME[S]U USE ME reversed
49 Gets on height on headland producing state of invigoration (9)
F[A]RESHNESS FARES (Gets) + H (height) + NESS (headland)
50 Con ship on river (4)
S[S]CAM SS (ship) + CAM (river)
1 Wasted uncle Ted takes acid (one for a bit of dope) (9)
EM[E]ACIATED EME (Uncle) + TED containing ACIA (ACID with A ie one replaced by D ie first letter of DOPE)
2 Lines of police cars avoiding slipperiness? (6)
3 Letter found in Japanese [square (TWENTY-FIVE)] (4)
ETA Two meanings
4 Moira's blown her top after cut drug (7)
PARE[O]IRA (M)OIRA following PARE (cut)
5 Strategic Health Authority has destroyed [crew (EIGHT)] (4, abbrev)
SHA Anagram of HAS
6 Stock all correct from tax region (7)
TALO[N]OKA TALON (Stock) + OK (all correct) + A (from)
7 Hoax doctorate? I'm not convinced! (5)
HUMPH[D] HUM (Hoax) +PHD (doctorate)
8 [(TWELVE) Apostles] accepted New Testament soldier, perhaps (4)
ANT A (Accepted) + NT (New Testament)
9 Cat found in filthy place, it's cheap and smoky (5)
ST[M]OGY MOGT (Cat) inside STY (filthy place)
10 Ate bark with a bit of liquorish making jaw painful (7)
[A]TETANAL ATE + TAN (bark) + A + L (first letter of LIQUORICE)
12 Something delightful put down as something valuable (5)
[G]ASSET GAS (Something delightful) + SET (put down)
20 [Tea for (TWO)] mother earth with wise man (5)
MAGE MA (mother) + GE (earth)
21 [Former Court of Session (FIFTEEN)] approval for fortune (5)
SEAL Two meanings
22 Fern seen in [February (TWENTY-EIGHT)] at a ramble (5)
TARA Hidden in aT A RAmble
23 Instrument demonstrating elide and strum creatively (9, two words)
27 One who wins over England with Grace batting (7)
ENGAGER[C] ENG (England) + AGERC (anagram of GRACE)
29 Criticise spread of a splendid palace (7)
[S]LATERAN SLATE (Criticise) + RAN (spread)
30 One leaving Iraq with Great Britain as Great Satan initially creates lots of confusion (7)
RA[Q]GBAGS IRAQ less I (one) + GB (Great Britain) + AGS (first letters of As Great Satan)
34 In an odd embrace? (5)
35 Fruit and water provide clear protection for pupil (6)
[A]CORNEA ACORN (Fruit) + EA (water)
36 Ale's unfinished, beer notwithstanding (5)
ALBEE[R] ALE, less last letter, + BEER
37 Type of Romanian hovel a Christian houses (5)
V[E]LACH Hidden in hoVEL A CHristian
40 Other people's [adult entertainment (EIGHTEEN)] operations (4)
OPS Two meanings
42 Be happy with average [catch (TWENTY-TWO) (4)
AVE Two meanings
44 I'm here, [high (FIVE)] and mighty, prince declares finally (4)
YES Last letters of mightY princE declareS