One of the joys of solving Inquisitors is learning about the stories behind some of the quirky themes that setters use. Cranberry’s puzzle is an excellent example of scientific theory giving rise to an interesting practice.
The preamble was of medium length and told us that a service provider is making cuts, with efficiency in mind. The theme (7,9,7), provided in part by the unclued entry and confirmed by its positioning, suggests possible outcomes. Each of twelve thematically relevant clues contains an extra word; in order, the first and last letters of extra words spell an alternative description. To illustrate one possible outcome, draw three straight lines, covering: the unclued entry; the ervice provider; and, what the service provider is doing.
I thought this was a fairly challenging solve as Cranberry didn’t hold back on the intricacy of some of the wordplay or on the introduction of the names of less well known artists. writers and musical instruments.
I made steady progress through the solve but took a while to make sense of the letters building up in the middle row. I stared at P_L G___LN___ERS for a long time before I deduced that three of the early extra words in the down clues could generate there word NUMBER which could also be part of a word in the middle row. I studied a course in number theory in mathematics at University many moons ago and somehow managed to dredge up a memory of POLYGONAL NUMBERS which fitted the letters I had already in the middle. A bit of googling revealed that the full term for the theme is CENTRAL POLYGONAL NUMBERS (7,9,7) thereby meeting the requirement of the preamble.
I wasn’t having much success in identifying the extra words in the across clues as the ideas I had didn’t generate a sensible word or words. The only extra word I was sure of was Iowa in 16 across. I had ideas about ‘neighbours‘ in 1 and ‘Tucson‘ in 42a, neither of which were correct. I was, though, having more success with the extra words in the down clues and could see a likely NUMBERS PLUS ONE forming at the end of the alternative description. A bit more research showed that CENTRAL POLYGONAL NUMBERS can also be referred to as TRIANGULAR NUMBERS PLUS ONE. With a bit of reverse engineering I was then able to locate all the extra words in the across and down clues as shown in the table below.
| Clue Number | Extra Word | Letters |
| Across | ||
| 1 | TeenageR | TR |
| 16 | IowA | IA |
| 37 | NegotiatinG | NG |
| 46 | UnoriginaL | UL |
| Down | ||
| 1 | AfteR | AR |
| 2 | NouveaU | NU |
| 4 | MoB | MB |
| 7 | EditoR | ER |
| 11 | SwaP | SP |
| 16 | LoU | LU |
| 22 | SO | SO |
| 34 | NewcastlE | NE |
generating TRIANGULAR NUMBERS PLUS ONE
I know a little bit about TRIANGULAR NUMBERS as they feature frequently in number puzzles found in The Magpie crossword magazine and in The Listener series. After writing down the series of TRIANGULAR NUMBERS as shown below, I realised that all the clues providing the extra words were all 1 more than a TRIANGULAR NUMBER, thus illustrating the alternative description of the theme.
| Triangular numbers | 0 | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 |
| Thematic clue numbers | 1,1 | 2 | 4 | 7 | 11 | 16,16 | 22 | 29 | 37 | 46 |
The next breakthrough came during a bit more research into CENTRAL POLYGONAL NUMBERS where I came across something called the LAZY CATERER‘s sequence which identified the maximum number of pieces of PIZZA that could be produced using only straight line cuts in a circular PIZZA. The cuts will give you more pieces of PIZZA than you might expect from symmetrically placed cuts, but you will almost certainly get howls of anguish from a few recipients of the pieces of PIZZA as some will be very small while others may be quite large.
I could see SLICING PIZZAS in a straight line in the grid and a further search revealed LAZY CATERER running NE in a straight line from the L of LEATHERED (44a) to the R of WAGGONERS (10a)
In summary, we have the three straight lines, one covering the unclued entry [forming part of the theme] POLYGONAL NUMBERS, one covering the service provider LAZY CATERER and one covering what the service provider is doing SLICING PIZZAS
There was some splendid misdirection in the title and preamble provided by Cranberry. Given the title Circle Line / Central Line I wonder how many of thought of London Underground as the service provider or though of a Utility company. Certainly a PIZZA house wasn’t in my immediate thoughts for being part of the endgame. Also I suspect the phrase ‘making cuts with efficiency in mind’ raised thoughts about reducing staff numbers.
I thought the surfaces of the clues and the wordplay was high quality such that I have had to into quite a bit of detail in the table below to illustrate the parsing.
I found an interesting video that bring all the pieces of them together which can be viewed here
I have given two grids below, the first shows the filled grid before highlighting the lines (cuts). the second shows the lines (cuts) and indicates the 7 pieces of PIZZA that will be generated by the three cuts.
The title Circle Line / Central Line refers to a PIZZA (circle) and the unclued central row (line).
Despite the mathematics telling me I can get more slices than I would think from a circular PIZZA, I’m not sure that displaying the name LAZY CATERER on the front of the shop would help me build a thriving business.
Thanks to Cranberry for a splendid puzzle.
| No | Detail, including amended clues at the thematic clue numbers | Extra Word |
| Acro34ss | ||
| 1 | Rogue teenager vandalised cars … are neighbours last to tell? (6)
Rogue vandalised cars … are neighbours last to tell? (6) RASCAL (rogue) Anagram of (vandalised) CARS + A (are [metric unit of area measurement] + L (final letter of [last to] telL) – ‘neighbours’ seems to be just an additional word to improve the surface of the clue, but it also indicates that RASC* and A are close to [neighbours] L RASC* A L |
TeenageR |
| 5 | As lover once told you, Italy’s changed – you’ll never see it in such a state! (10)
ASEXUALITY (state of being without sex – the clue suggests you won’t find Italians going without sex) AS + EX (former [once] partner or lover) + U (sounds like [told] YOU) + an anagram of (changed) ITALY AS EX U ALITY* |
|
| 10 | Drivers with a group of Arsenal fans, one nil down (9)
WAGGONERS (people who drive WAGONs; drivers) W (with) + A + G (group as in the G7, the Group of Seven, an informal forum of seven of the world’s advanced industrialised democracies) + GOoNERS (Arsenal fans) excluding (down) one of the Os (character representing zero [nil]) W A G GONERS – either O can be the one omitted |
|
| 12 | Reportedly leaves singing groups (6)
CHOIRS (singing groups) CHOIRS (sounds like [reportedly] QUIRES [sheets {leaves} of paper]) CHOIRS |
|
| 13 | Strange meteoroid occasionally seen close to Earth (5)
EERIE (strange) EERI (letters 2, 4, 6 and 8 [occasionally seen] of mEtEoRoId) + E (Earth) – ‘close to’ is another phrase helping to improve the surface and position component parts of the wordplay together EERI E |
|
| 16 | Not so cloudy, Iowa valley turns cooler (3-3)
Not so cloudy, valley turns cooler (3-3) AIR-CON (AIR-CONditioner [ a cooler]) (NO [not so] + C [cloudy] + RIA [a normal drowned valley]) all reversed (turns) (AIR C ON)< |
IowA |
| 18 | Deliberately oldfangled transport system – making money the focus of governors (5)
RETRO (design, style, etc which deliberately reverts to or recreates the past) METRO (transport system) with M (money as used in M1, M2 categories of money supply in the UK) replaced by (making [M into]) R (central letter of [focus of] goveRnors) R ETRO |
|
| 19 | Installed trench, about to get pipe down (8, 3 words)
TURN IT UP (stop [saying] it; pipe down) (PUT IN [installed] + RUT [trench]) all reversed (about) (TUR N I T UP)< |
|
| 21 | Gut feeling ultimately bad (4)
GILL (to gut [fish]) G (last letter of [ultimately] feelinG) + ILL (bad) G ILL |
|
| 23 | Oil covering cheeks of backside (4)
OTTO (a very fragrant essential oil made in Bulgaria and elsewhere, chiefly from the damask rose [found under attar in Chambers]) bOTTOm (backside) excluding (covering) the outer letters (cheeks) B and M OTTO |
|
| 24 | Barrel nearly tipped over – oddly, four litres is as much as it’ll hold (6)
TUBFUL (as much as a tub [could be a barrel] will hold) BUTt (barrel) excluding the final letter (nearly) T, then reversed (tipped over) + FU (letters 1 and 3 [oddly] of FoUr) + L (litres) TUB< FU L |
|
| 28 | I’m ready to go on radio to extend influence (6)
IMPACT ([extend] influence) I’M + PACT (sounds like [on radio] PACKED [ready to go]) IM PACT |
|
| 33 | Fold over Rolling Stone (4)
OPAL (semi-precious gemstone) (LAP [fold] + O [over]) all reversed (rolling) (O PAL)< |
|
| 35 | Somewhere to eat around outskirts of Florence (4)
CAFE (somewhere to eat) CA (circa; around) + FE (outer letters of [outskirts of] FlorencE) CA FE |
|
| 37 | Director acquires drug, negotiating round back of studios (8)
Director acquires drug, round back of studios (8) SCORSESE (reference Martin SCORSESE [born 1942], American film director) (SCORES [acquires] + E [ecstasy; drug]) containing (round) S (last letter of [back of] studioS) SCOR (S) ES E |
NegotiatinG |
| 39 | Finish off presentation in time for Germans playing Russians (5)
ZENIT (reference the football team [players] ZENIT St Petersburg, a Russian side; playing Russians) N (last letter of [finish off] presentatioN) contained in (in) ZEIT (German for time) ZE (N) IT |
|
| 41 | Places to stay? Most wanted the Spanish sun (6)
HOTELS (places to stay) HOT (sought after; most wanted) + EL (Spanish for ‘the’) + S (sun) HOT EL S |
|
| 42 | Shopping centre in Tucson, Arizona, with public library area outside (5)
PLAZA (American [Tucson] term for a shopping centre) AZ (Arizona) contained in (with … outside) (PL [Public Library] + A [area]) PL (AZ) A |
|
| 43 | International community inspired by quiet man, free spirit (6)
PNEUMA (soul; spirit) EU (European Union, an international community) contained in (inspired by) (P [piano; quiet] + an anagram of [free] MAN]) P N (EU) MA* |
|
| 44 | Libertine’s clothing articles, bloodstained, taken to the cleaners (9)
LEATHERED (beaten by a large amount; taken to the cleaners) LE (outer letters of [clothing] LibertinE) + (A [indefinite article] + THE [definite article] giving ‘articles’) RED (bloodstained) LE A THE RED |
|
| 45 | Take on board assorted items and sail away, renouncing established routine (10)
ASSIMILATE (take fully into the mind; take on board) Anagram of (assorted) ITEMS and SAIL and Away excluding (renouncing) WAY (established routine) ASSIMILATE* |
|
| 46 | Dull in the head, unoriginal but cunning (6)
Dull in the head, but cunning (6) BLEARY (dim; dull) B (first letter of [in the head] But + LEARY (alternative spelling of leery [cunning]) B LEARY |
UnoriginaL |
| Down | ||
| 1 | Edited perhaps, after reading we served up meaningless repetition (7)
Edited perhaps, reading we served up meaningless repetition (7) REWROTE (edited? – perhaps a bit beyond editing if the work has been rewritten) R (reading – as in one the three Rs [reading; [w]riting and [a]rithmetic]) + WE reversed (served up; down entry) + ROTE (meaningless repetition) R EW< ROTE |
AfteR |
| 2 | Before a drop of Aussie nouveau, maybe Alan Partridge drained veg (8, 2 words)
Before a drop of Aussie, maybe Alan Partridge drained veg (8, 2 words) SUGAR PEA (mange tout [vegetable]) SUGAR (reference Lord Alan SUGAR [born 1947]; example of a person named Alan) + PE (letters remaining in PartridgeE when the central letters are removed [drained] + A (first letter of [drop of] Aussie)]) SUGAR PE A |
NouveaU |
| 3 | Most suspicious habit of fighting when getting plastered? (7)
CAGIEST (most wary; most suspicious) GIE (judo or karate costume; habit [customary dress] of fighting) contained in (when getting [covered by]) a plaster CAST – therefore getting plastered) CA (GIE) ST |
|
| 4 | Somewhat critical of mob turning up (5)
Somewhat critical of turning up (5) ALOFT (on high; up) ALOFT (hidden word [somewhat] in criticAL OF Turning) ALOFT |
MoB |
| 5 | Accepted my new socioeconomic classification (5)
ACORN (a Classification of Residential Neighbourhoods, a type of socio-economic classification that identifies the area and housing in which people live and can be used for direct mail or market research) A (accepted) + COR (gosh! my!) + N (new) A COR N |
|
| 6 | Cutting ends off seasonal decoration (7)
SLICING (cutting) SL (outer letters of [ends off] SeasonaL) + ICING ([cake] decoration) SL ICING |
|
| 7 | Current editor discovered here, amongst books still (5)
Current discovered here, amongst books still (5) INERT (immobile; still) I (electric current) + (ER [letters remaining in hERe when the outer letters H and E are removed {dis-covered}] contained in [among] NT [New Testament {books}]) I N (ER) T |
EditoR |
| 8 | Every now and then these old fellows make the first move (6, 2 words)
TEE OFF (make the first move) TEE (letters 1, 3 and 5 [every now then] of ThEsE) + O (old) + (F [fellow] + F [fellow] giving fellows) TEE O F F |
|
| 9 | King’s wearing this ridiculously loose sleeved garment (5)
SHIRT (loose sleeved garment) R (Rex; King) contained in (wearing) an anagram of (ridiculously) THIS SHI (R) T* |
|
| 11 | One assumes straightforward swap, sacrificing pawn, as easy as pie? (6)
One assumes straightforward, sacrificing pawn, as easy as pie? (6) SIMILE (‘as easy as pie’ is an example of a SIMILE) I (Roman numeral for one) contained in (assumes) SIMpLE (straightforward) excluding (sacrificing) P (pawn) SIM (I) LE |
SwaP |
| 14 | Saucy stuff in tabloid, suitable for kids? (4)
RAGU (in Italian cookery, a meat and tomato sauce) RAG (an informal term for a tabloid newspaper) + U (classification indicating that a film is suitable for viewing by children) RAG U |
|
| 15 | Beasts regularly consuming large bendy plant (8)
EELGRASS (a grasslike flowering plant of the pondweed family, growing in seawater) ESS (letters 2, 4 and 6 [regularly] of bEaStS) containing (consuming) an anagram of (bendy) LARGE E (ELGRA*) SS |
|
| 16 | Music producer Lou Adler’s first no. 1 in US, with incredibly hot European artist (8)
Music producer Adler’s first no. 1 in US, with incredibly hot European artist (8) AUTOHARP (a kind of zither, with button-controlled dampers, which produces chords; music producer) A (initial letter of [first] Adler) + U (first letter of [number one in) Us + an anagram of (incredibly) HOT + ARP (reference Hans ARP [1886 – 1966], German-French artist and sculptor) A U TOH* ARP |
LoU |
| 17 | Irritation – scratching is making little difference (6)
NUANCE (delicate or subtle degree or shade of difference) NUisANCE (irritation) excluding (scratching) IS NUANCE |
|
| 19 | Subject of detailed cover photo (5)
TOPIC (subject) TOp (cover) excluding the final letter (de-tailed) P + PIC (PICture; photo) TO PIC |
|
| 20 | Picking up posh menu, wanting starter – check for one without nuts (6)
EUNUCH (castrated man, one without testicles [nuts]) (mENU excluding the first letter [wanting starter] M, containing [picking up] U [upper-class; posh]) + CH (check) E (U) NU CH |
|
| 22 | Yarn from backward yokel’s so ill-judged (5)
Yarn from backward yokel’s ill-judged (5) LISLE (a long-stapled, hard-twisted cotton yarn) LISLE (reversed [backward] hidden word in [from] yokEL’S ILl-judged) LISLE< |
SO |
| 25 | Entering quiet period, though Ed’s bounding with energy (6)
LIFULL Edmund Spenser’s [Ed’s] word for ‘full of vital energy’; bounding with energy) IF (though) contained in (entering) LULL (quiet period) L (IF) ULL |
|
| 26 | Climber taking week off following onset of lung complaint (8)
LISTERIA (a bacterium frequently found in certain foods [especially chicken, soft cheeses, etc], which if not killed in cooking can affect the central nervous system) L (first letter of [onset of] Lung) + wISTERIA (climbing plant [climber] excluding [taking … off] W [week]) L ISTERIA |
|
| 27 | Uncommonly ignorant preacher, Mexican, claiming magical influence over time (7)
MARTEXT (an ignorant preacher) (MEX [MEXican] containing [claiming] ART [magical influence]) + T (time) M (ART) EX T |
|
| 29 | Big Newcastle game reorganised (4)
Big game reorganised (4) MEGA (big) Anagram of (reorganised) GAME MEGA* |
NewcastlE |
| 30 | Special deals, amazing – essentially next to nothing on return fare from Naples (6)
PIZZAS (fare from Naples, said to be the home of the PIZZA) (S [special] + A [central letter of {essentially} deAls] + Z [central letter of {essentially} amaZing] + ZIP [informal term for nothing]) all reversed (on return) (PIZ Z A S)< |
|
| 31 | Ship’s engineers supporting voyage being cut short (7)
TRIREME (ancient Greek galley; ship) TRIp (voyage) excluding the final letter (cut short) P + REME (Royal Electrical and Mechanical Engineers) TRI REME |
|
| 32 | A most prosperous time indeed after number on dole reduced by 75% (7)
NOONDAY (figuratively, the time of greatest prosperity) NO. (number) + ON + DOLE excluding the last three of the four letters [75% reduced]) + AY (yes; indeed) NO ON D AY |
|
| 34 | Sole proprietor expected less mind-numbing admin to begin with (5)
PELMA (sole [of the foot]) PELMA (first letter of [to begin with] each of Proprietor, Expected, Less, Mind and Admin) P E L M A |
|
| 36 | Poet briefly stayed in failed state (5)
YEATS (reference W B YEATS [1865 – 1939], Irish poet) Anagram of (in failed state) STAYEd excluding the final letter (briefly) D YEATS* |
|
| 38 | German author expressed doubt about his Sämtliche Werke, finally? On the contrary! (5)
SEUME (reference Johann SEUME [1763 – 1810], German author) SEE (last letters [ultimately] of each of hiS, sämtlichE and werkE) containing (about) UM (expression of doubt) – this construction is the opposite of that indicated by the first part of the clue that would lead to UM containing SEE – on the contrary) SE (UM) E |
|
| 40 | Daniel’s bad deal leaving him with no money left, nothing (5)
NIHIL (nothing) daNIel excluding (leaving) an anagram of (bad) DEAL + HIm excluding (with no) M (money as in M1, M2 etc, UK money supply categories) + L (left) NI HI L |
This makes two unusual grid shapes in a row, and this one was considerably larger. I thoroughly enjoyed the solving phase of this puzzle and appreciated the skill and imagination that went into writing the clues. It was interesting, and amusing, to then find that the message formed from the twelve pairs of letters (‘triangular numbers plus one’) led me straight back to the clues that yielded those letters!
I guessed first that the required phrase was ‘centred triangular numbers’, which are 1, 4, 10, 19, 31 and 46, …, which evidently had no significance here (although 1 and 46 happen to be in the series of ‘triangular numbers plus one’).
Although SLICING PIZZAS was there in plain sight, it had no significance for me, and not knowing what I was meant to find I did not see LAZY CATERER (or more likely saw no significance in it). I wanted the line through POLYGONAL NUMBERS to be one side of a triangle, but I was obviously mistaken with that idea.
Thanks to Cranberry for the excellent clues (and the clever design) and to duncanshiell for the clear and expert blog.
Being unhampered by any prior knowledge of central polygonal numbers or triangular numbers, but making some word guesses after solving most of the clues, I rambled around on Google and duly bumped into the Lazy Caterer. I still don’t understand the mathematical concepts, but completing the puzzle then became pretty straightforward (though I admire anybody who recognised Johann Seume without a check!). Like Duncan I needed quite a bit of reverse engineering to get to Triangular, having inaccurately identified some superfluous words (while assuming that Iowa was necessary to indicate that RIA was American, even though it isn’t). I don’t think Cranberry would last long in a pizza parlour if the completed puzzle is his best shot at a fair division of the product, but I enjoyed it as a crossword.
I’m afraid that for me it was too much of a leap of the brain from ‘CENTRAL POLYGONAL NUMBERS’ to pizzas unless you wanted to spend hours reading Google. I entered the phrase and the Wiki entry (which takes you to ‘CENTRED…’) says zilch about pizzas. Why would the average solver who knows nothing of the concept then go elsewhere to look for anything about pizzas?
I have to confess to not being well-disposed to the endgame as I thought there were one or two ambiguities re the ‘redundant’ words (1a being a good example) which caused me to take way too long than I should have deducing the phrase
Phew, what a scorcher! Lots of thanks to Cranberry and duncanshiell. POLYGONAL NUMBERS stirred my recollections of a great many Scientific American pop-maths columns, and Googling the revealed number sequence gave “Lazy caterer’s sequence” (at Wikipedia) in the first page of hits, so it all seemed quite fair — especially since pizza, already noticed in the grid, appears in the first sentence of that article.
@2 SEUME was the very last answer I completed, having inferred the name from the wordplay but failed to find him in the one-volume encyclopedia or biographical dictionaries on our dining-room reference shelf. I was duly relieved by another Google search.
Rectangular grids… so old hat. I enjoyed the solve, but when I suspected a mathematical or research heavy endgame, I retired. Enjoyed the video, though. Thanks to Cranberry and duncanshiell for a very helpful blog.
@David Langford
I suppose it’s ‘fair’ if you consider an in-depth knowledge of what triangular numbers are and therefore could deduce a number sequence, but it may come as a surprise to you (and the editors of the Inqy and Listener) that some of us are not mathematicians and have zero interest in googling miscellaneous maths pages of Wiki…
Another puzzle where my personal tolerance ratio of gridfill/endgame was way off the scale.
I’m no mathematician, but found the puzzle perfectly accessible and solvable with a couple of minutes asking Bing (not Google) about central polygonal numbers, and thereby discovering the lazy caterer idea. I still don’t feel I know what these numbers are, or why they are interesting, or why the caterer is lazy, but none of that seemed necessary to get to grips with clue solving or pizza slicing, (though I’d agree with Bingy that it wasn’t always easy to identify redundant words in the Across clues, because there seemed to be several potential candidates). Thanks to Cranberry and Duncan.
I thought this was an excellent construction with several layers unfolding smoothly. The grid filling was challenging but enjoyable, and it became clear which were the thematically relevant clues. I then quickly came across somewhere on Google a link between central polygonal numbers and “lazy” caterers dividing up polygonal items with a minimum number of cuts (if unfairly), which matched the rubric about a service provider making cuts. And though I hadn’t found pizzas specifically cited, “slicing” was a straightforward spot in the grid with pizzas following it. All in all a very satisfying solve. Many thanks to Cranberry and duncanshiell.
Thoroughly enjoyed this! In contrast to some other commenters I did not find the ‘mathematical’ aspect of the endgame remotely off-putting, despite having zero aptitude or even interest in maths… in fact I still don’t quite understand it 😁 but once the grid was filled the lines SLICING PIZZAS and LAZY CATERER jumped out and a quick Google confirmed it. So in fact this was a far less arduous endgame than many other IQs, I thought. Like some others a little back-parsing and message gap-filling was needed here and there. But overall, good fun.
Big thanks to Cranberry and Duncan.
Not being of a mathematical bent didn’t help, but I found Polygonal Numbers, Triangular Numbers Plus One (and where they were), and even spotted that the service provider could be a caterer (I didn’t spot lazy). But there was nothing to point me to Central (or maybe I was following the wrong Google track). So whilst I had fun with the solving and finding the extras, I didn’t get a slice, so 7/10 – which is better than I ever did in mathe class.
As usual the preamble was confusing at first so we did our normal trick of just trying to solve the clues regardless.
Joyce had heard of POLYGONAL NUMBERS but couldn’t remember anything specific about them so it needed googling.
The first search revealed the LAZY CATERER which made things easier. It’s a strange connection to base something on a person who cannot locate the centre of a circle when they are cutting a pizza though.
All in all, an enjoyable puzzle so thanks to Cranberry and Duncan.
Rather late to the pizza party.
Unlike quite a few of the others, I am of a mathematical bent but didn’t find that at all helpful! I’d come across the problem of dividing a convex shape into as many pieces as possible with the fewest straight lines but had never heard of the Lazy Caterer, or indeed central polygonal numbers.
I found it a bit odd that the thematic relevance of the clue numbers that led to TRIANGULAR NUMBERS PLUS ONE was that they were triangular numbers plus one – that didn’t really add anything to what we already knew. Maybe it was just some sort of confirmation.
Anyway, thanks Duncan for the blog & thanks Cranberry for the introduction to polygonal numbers.
HolyGhost @ 13
I think the clue numbers do have a thematic relevance beyond being one more than triangular numbers.
The clue numbers in ascending order represent the maximum number of distinct pieces you can get from 0, 1, 2, 3, 4…. straight line cuts of a pizza
0 cuts leaves 1 piece, 1 cut will produce 2 pieces, 2 cuts can produce 4 pieces, 3 cuts can produce 7 pieces as illustrated in the blog, 4 cuts can generate 11 pieces etc …
Duncan @14: had the phrase emerged as something like MOST PIECES BY STRAIGHT CUTS then the clue numbers would indeed have added something. But what we had here was more akin to the phrase PRIME NUMBERS being generated by clues 2, 3, 5, 7, 11, 13 etc.