The final answer is the name of a fictional character.
Image credits:
7: Rama, cc by-sa 2.0 fr | 11: Jonathunder, GFDL 1.2 | 13: Volatus, cc by-sa 3.0 | 15.1: ZngZng, cc by-sa 3.0 | 15.2: Ian Kirk, cc by-sa 2.0 | 16: Antonio Litterio, cc by-sa 3.0 | 20: J. Patrick Fischer, cc by-sa 3.0 | 21: Kneiphof & historicair, cc by-sa 3.0 | 22 & 25.2: Jeff Dahl, cc by-sa 4.0 | 23: Mohandoss Sampath, cc by-sa 3.0 | 24: Liné1, cc by-sa 4.0
I'd wager we are looking at
a rebus. Naturally. :-)
Start by unscrambling the image:
Assuming that all the circle segments in the corners and on the sides of the images combine to make single-coloured full circles, there's only one way to assemble the jigsaw puzzle into a 5x5 grid as required by the title:
Then, solve it row by row
Perpendicular - Pound - Empty set - Jewish (Hebrew) Plus - Dollar
Naira - Katakana Yo - Right Semijoin - Gimel - Pilcrow
Rho - Scruple - Tshe - Universal quantifier - Vertical Bar
Small Eth - Existential quantifier - Small Capital Lambda - Exclamation - Shang
Open O - Three - Up Tack - Schwa - Latin Alpha
This is very interesting:
These all are unicode characters of some kind. Particularly of the kind that look like upside-down letters. Write them down, stand on your head, and read.
Like so: (All polished up with a little help from OP's comments below)
⟂ £ ∅ ﬩ $ ₦ ヨ ⋊ ℷ ¶ ρ ℈ ћ ∀ | ð ∃ ᴧ ! 上 ɔ 3 ⊥ ə Ɑ
With a little squinting, it reads
"Detective played by Ken Stott", which finally brings us to the answer.
Appendix: Full reasoning for the individual images, in the unscrambled order:
1:
Pe
er to peer, pen, dick, Ural, swap r and l to get ular
2:
a pound is equal to 453.6 grams.
3.
A set with no elements is called an empty set
4.
Star of David hints at Hebrew; it is the symbol for Judaism. The Hebrew language has an alternative symbol for plus.
5.
Doll + upside-down (reversed) Egyptian god Ra = ar
6.
Latter part of Rastafarian reversed = Naira, the Nigerian monetary unit.
7.
Katana + "ka" in the middle = katakana, one of the Japanese writing systems. Half of Yo-yo is yo, a Japanese syllable.
8.
Arrow pointing to the right, followed by New York Times, reversed, for semi, and join
t.
9.
An upside-down (reversed) lemming, with the letters N and M removed: g
nimmel
10.
Pill + Crow, slightly misspelled.
11.
Rh
ino
12.
"ESC" minus E plus ruble with a letter exchanged: "ruple"
13.
The Egyptian god Seth, with letters scrambled in the order given, to get "Tshe". (Surprisingly, didn't have anything to do with the numbers being an approximation for $1000 \times \pi$)
14.
Universal serial bus, "quid pro quo" - d + ant in the middle = quanti, Semper fi, reconquista (A historical period on the Iberian peninsula) reversed for er
15.
Quite simply vertically written word bar
16.
S means small when written on a tee shirt, eth fits all the empty spaces, creating an English word each time.
17.
exit + S = exist, vial with the letter swap = ential, quant
ummechanics, wifi, flag of Eritrea, country code er.
18.
Ottawa is the (small) capital of Canada, lamb, Cyrillic word "da"
19.
Ex libris, clam, AT&T, hydrogen ion
20.
Picture of Shanghai, with the first character of the city's Chinese name hidden
21.
o, pen, o
22.
tree with an added h gives three
23.
Arrow pointing up, t
rack
24.
Flag of Germany hints at language, the colour pointed at is black, which is Schwarz in German, picking the first 5 of the 7 letters gives Schwa.
25.
(Original puzzle had an error, which is still visible in the unshuffled image; instead of "Si", it was supposed to read "Sn".) Map has Los Angeles highlighted for LA, Sn is the symbol for the element tin, and the letter is of course an alpha.
A rearrangement.
The 25 pieces can be rearranged into 4 groups using the coloured circles as cues, as this crude edit shows.
Where to go from here?
Some partials. Combine with @Weather Vane's answer to put them in the correct order for reading.
If we assume that in the OP they are numbered first row 1-5, second row 6-10, etc., then we have:
1.
THREE = TREE + H
2.
EXISTENTIAL = EXIT + S + VIAL (with V = ENT) flag of ERITREA
3.
NULL or NULL SET or EMPTY?
5.
? = Silicon + alpha + LA
6.
SEMIJOIN = IMES (reversed) + JOINT - T
7.
? = KA + KATANA + YO (half of YOYO)
8.
SETH, STETH, ? = the letters that fill in the blanks are ETH, and there's an S in a T-shirt
11.
RHO = RHINO - IN
13.
U = capital (Ottawa) ewe
14.
QUANTIFIER = U + ANT + QUID - D + FI + ER (RE reversed)
17.
POUND (453.6 grams is approximately 1 pound)
18.
SCRUPLE = ESC - E + RUBLE (with B = P)
21.
PERPENDICULAR = PEER - E + PEN + DICK - K + URAL (with L and R switched)
22.
TSHE = SETH rearranged to take in order 3142
23.
TACK = TRACK - R
24.
EXCLAMATION = EX + CLAM + AT + ION
25.
DOLLAR = DOLL + AR (god RA upside down)
Partial answer
Continuing from Weather Vane's observation
that the squares can be rearranged into four groups if we require that the quarter-circles at some of their corners match up,
in fact
those four groups fit in a unique way into the original 5x5 square shape. Some of the rebuses are easier to make sense of than others; so far I have the following:
perpendicular pound empty-set ??star+?? ??doll??
or katakana-yo right-semijoin ??? ??pilcrow??
rho scruple ??combination-set?? quantifier vertical-bar
seth existential-qualifier capital-lambda exclamation shang
??open-o?? three ??ail?? schwa ??Si-alpha-los-angeles??
(Credit to Bass for noticing that it's specifically a right semijoin, which I hadn't spotted.)
I haven't given detailed explanations of how the rebuses work because I think they're mostly easy to see once you have the answer, but here are a couple of the subtler ones:
"schwa" is because we have the German flag, we're pointing to its black stripe, hence schwarz (German for black), and we want 5/7 of this, making schwa. "katakana yo" is because "ka" has been inserted into the middle of a katana (plus half a yo-yo). "or" is half an Afro, reversed. I don't actually believe "seth" since it doesn't seem to be the name of a symbol, but the point is that you can insert "eth" into all those gaps. "Shang" is because we have a picture of some towers in Shanghai, plus a gap followed by the "hai" character.
which makes me think that
each square is leading to a single symbol, something like this:
⟂£∅??
∨彐⋊?¶
ρ℈??|
?∃Λ!上
ɔ3?Ə?
shoover's answer has some plausible explanations for a few of my gaps, yielding
⟂£∅?$
∨彐⋊?¶
ρ℈Ћ?|
?∃Λ!上
ɔ3⊣Ə?
It isn't entirely clear to me what to do next;
it looks a little as if something good might happen if we read the whole thing upside down, though. (It looks a little like a word-ladder with only one letter changing at each step, but so far I can't make that work and I think it's probably wrong.) At any rate it's striking how many upside-down-E-like things there are in the second column. Perhaps the "or" symbol is actually meant to be a vertical bar and we're meant to read the pilcrow, even upside down, as a P, in which case that row yields "pixel". I suspect the open-o may be meant to be read as a D, even though to my mind that only works if we don't invert it. It's possible that the fourth row might say "fiver", in which case maybe I can pretend that the second row says "hazel" and there's a Watership Down reference going on. But so far this is all just speculation.
Ah!
I bet it says DETECTIVE - LIVED BAKER STREET, in which case the answer is SHERLOCK HOLMES. (Credit where due: this did not occur to me until I saw Bass's inspired guess that in the mysterious Los Angeles box "Si" is a typo for "Sn", so that that square contains an "a".) If this is right, then clearly I have entirely the wrong symbol in the middle of the first line (an empty set) since it looks nothing like "re" or "ee".
