The Genius is the Guardian’s monthly puzzle targeted at solvers who like to have an endgame of some sort over and above the core task of solving clues. The puzzles are often a means of learning more about subjects where solvers may just have scratched the surface in the past.
This month’s puzzle was educational.
The preamble told us that “In all but eight clues, wordplay omits a letter. In order, these suggest how the remaining clues must each have a word removed before solving. Solvers must complete the puzzle with the unclued entry which might represent the starting point of the theme”.
The crossword is based around the Fibonacci series, which starts 0,1 and then follows a systematic pattern:
0,1,1,2,3,5,8,13,21,34,55 … where xn = xn-1 + xn-2
The message spelt out from the omitted letters is
FIBONACCI SEQUENCE NUMBERS, which directs us to look at clues whose number lies in the sequence.
The table below shows what words should be removed from eight clues. The clues listed are Fibonacci numbers 1,2,3,4,8,13 and 21. Each word removed from these clues represents a thing that displays Fibonacci sequence characteristics, or is a word associated with Fibonacci himself.
As there is no clue numbered 0 (zero) in the puzzle, the unclued light is ZERO [9 down] to reflect the first number in the Fibonacci series.
| Clue | Word | Fibonacci association |
| 1a | spiral | The Fibonacci spiral: an approximation of the golden spiral created by drawing circular arcs connecting the opposite corners of squares in the Fibonacci tiling. |
| 13a | Leonardo | Leonardo Bonacci [circa 1170 – circa 1240/50, also known as Leonardo da Pisa, was best known as Fibonacci. |
| 21a | pineapple | A scientific study in 1970 found that most pineapples had 8-13-21 rows of fruitlets, with a few smaller ones at 5-8-13. It was suggested that a pineapple with more fruitlets for a given size would have a finer texture, and it was hoped that a pineapple with 13-21-34 rows could be found; however, the author of the study never found any pineapples exhibiting such a pattern. |
| 1d | sunflower | The Fibonacci sequence appears in a sunflower as opposing sets of spirals formed by its seeds, which always correspond to consecutive numbers in the mathematical sequence. |
| 2d | bees | In simple (i.e., non-rigorous) terms, each new generation of male bees has a number of ancestors that is the sum of the two preceding generations, matching the exact definition of the Fibonacci sequence. |
| 3d | rabbits | The Fibonacci sequence first appears in the book Liber Abaci (1202) by Leonardo of Pisa, [see above], known as Fibonacci. Fibonacci considers the growth of an idealised (biologically unrealistic) rabbit population and argues that it will follow his sequence. |
| 5d | Nautilus | Nautilus shells display a logarithmic spiral pattern that closely approximates a Fibonacci spiral. |
| 8d | artichokes | Artichokes display the Fibonacci sequence and the Golden Ratio because their leaves (called bracts) grow in overlapping, counter-rotating spirals that match consecutive numbers from the sequence (0, 1, 1, 2, 3, 5, 8, 13, 21…) |
The information in the table has been gleaned from internet searches. Please note that I am not a specialist in number theory, and I am aware that there are several unstated assumptions underlying the statements in the table above, and some of the scenarios are unrealistic. The Wikipedia article https://en.wikipedia.org/wiki/Fibonacci_sequence is a useful starting point for anyone who wants to read more about the subject.
The finished grid looked like this:
The clues were clear and parsed cleanly. I didn’t always deduce the letter omitted from wordplay immediately, but I gradually built up the message to the point it became clear that the full message would be as highlighted above.
This was an enjoyable puzzle that expanded my knowledge of the natural occurrences of Fibonacci sequences. For anyone who does numeric puzzles in the Magpie or in the Listener series, knowledge of Fibonacci numbers is often useful when solving.
| No | Detail
Omitted letter shown in fuchsia |
Word/Letter |
| Across | Clue
Amended clue (where applicable) |
|
| 1 | Free flow round Channel Islands spirals powerfully to begin with (10)
Free flow round Channel Islands powerfully to begin with (10) EMANCIPATE (set free from legal, social or political restraint or bondage) EMANATE (flow from) containing (round) (CI [Channel Islands] + P [first letter of {to begin with} Powerfully]) EMAN (CI P) ATE |
spirals |
| 6 | Pose riddle (4)
SIFT (sieve; riddle) SIT (pose) SIFT |
F |
| 10 | Fix section of main grandstand (7)
INGRAIN (fix a dye firmly in) INGRAN (hidden word in [section of] maIN GRANstand) INGRAIN |
I |
| 11 | Wine finally imported by Polish liquor shop (7)
SHEBEEN (an illicit liquor shop) E (last letter of [finally] winE) contained in (imported by) SHEEN (rub; polish) SHE (B E) EN |
B |
| 12 | Fertile spot’s unaltered (5)
OASIS (fertile spot) AS IS (unaltered) O AS IS |
O |
| 13 | ‘Drink and act cool, Leonardo‘, he directed (9)
‘Drink and act cool’, he directed (9) CHOCOLATE (a drink) Anagram of (directed) ACT COOL HE CHOCOLATE* |
Leonardo |
| 14 | Large democracy adjoining American state (7)
INDIANA (State of the USA) INDIA (large democracy) + A (America; American) INDIA N A |
N |
| 16 | Insect hotel: termites welcomed! (6)
ELATER (a skipjack beetle; insect) ELTER (hidden word in [welcomed] hotEL TERites) ELATER |
A |
| 19 | Return a small pouch (3)
SAC (pouch) (A + S [small]) reversed (returned) (S A C)< |
C |
| 21 | Sauce in pineapple dish regularly generates bodily chemicals (6)
Sauce in dish regularly generates bodily chemicals (6) LIPIDS (chemicals found in bodily tissues) LIP (insolence; sauce) + IDS (letters 1, 3 and 5 [regularly] of In DiSh) LIP I D S |
pineapple |
| 22 | Hair treatment maybe withdrawn for young animal (4,3)
LION CUB (young animal) (BUN [hair{style}] + OIL [treatment for hair]) all reversed (withdrawn) (LIO N CUB)< |
C |
| 24 | Raced south to Rome every so often in wet weather (9)
RAINSTORM (wet weather) RAN (raced) + S (south) + TO + RM (letters 1 and 3 [every so often] of RoMe) RAIN S TO R M |
I |
| 26 | Be examined again on sources of revolutionary epistemic intellectual theory (5)
RESIT (be examined again) REIT (first letter of [sources of] each of Revolutionary, Epistemic, Intellectual and Theory) R E S I T |
S |
| 28 | Pastry chefs’ bowls (7)
BEAKERS (bowls) BAKERS (pastry chefs) BEAKERS |
E |
| 29 | One essentially dud song tanks! (7)
AQUARIA (tanks) A (one) + U (middle letter of [essentially] dUd) + ARIA (song) A Q U ARIA |
Q |
| 30 | Place of fabrication (4)
LIEU (place) LIE (a fabrication) LIEU |
U |
| 31 | Without protection from New Delhi’s latitude, starts to suffer sunburn (10)
SHIELDLESS (without protection) Anagram of (new) DELHI ‘S + L (latitude) + SS (first letters of [starts to] each of Suffer and Sunburn) SHIELD* L E S S |
E |
| Down | ||
| 1 | I trade sunflower oil all over the place for leader (9)
I trade oil all over the place for leader (9) EDITORIAL (an article in a newspaper written by an editor or leader writer) Anagram of (all over the place) I TRADE OIL EDITORIAL* |
sunflower |
| 2 | Protection from bees, for example, in Australian island (5)
Protection from, for example, in Australian island (5) AEGIS (protection) E.G. (for example) contained in (in) (A [Australian] + IS [island]) A (EG) IS |
bees |
| 3 | Spooner’s weasel-like creature dropped rabbits in piece of furniture (9)
Spooner’s weasel-like creature dropped in piece of furniture (9) COATSTAND (a piece of furniture) Reverend Spooner would pronounce COATSTAND as STOAT (weasel-like creature) CANNED (stopped; dropped) COATSTAND |
rabbits |
| 4 | Stride around academy while generating solutions (8)
PANACEAS (cures for all things; solutions) (PACE [stride] containing [around] A [academy]) + AS (while) PA (N A) CE AS |
N |
| 5 | Silver coin found in Nautilus equates to nothing (6)
Silver coin found in equates to nothing (6) TESTON (a name for various silver coins, originally bearing a king’s or duke’s head) TESTON (hidden word in [found] equaTES TO Nothing) TESTON |
Nautilus |
| 7 | I enlist a criminal firm (9)
INELASTIC (unyielding; firm) Anagram of (criminal) I ENLIST A INELASTI* C |
C |
| 8 | Preserve leaves of globe artichokes for flavour (5)
Preserve leaves of globe for flavour (5) TINGE (trace; flavour) TIN (put in a can; preserve) + GE (outer letters of [leaves] GlobE) TIN GE |
artichokes |
| 9 | Unclued Light
ZERO |
|
| 15 | Spare plaid cut off (9)
DUPLICATE (another of the same kind, possibly a spare one) Anagram of (off) PLAID CUT DUPLICAT* E |
E |
| 17 | Bird moving position of head in noisy inspection (4-5)
LOOK-ROUND (an inspection) ROOK (bird) with the first letter (head) moved to the end [moving position of] to form OOKR contained in (in) LOUD (noisy) L (OOKR) OU N D |
N |
| 18 | Critic perhaps covering the Baftas initially sent back negative reactions (9)
REBUTTALS (negative reactions) (SLATER [critic] containing [covering] TB [first letters of [initially] of each of The and Beatles]) all reversed (sent back) (RE (B U T) TALS)< |
U |
| 20 | Topeka balcony restricts rising American party (8)
CLAMBAKE (American or Canadian term for a picnic, often by the sea, at which CLAMs etc are BAKEd) CLAMBAKE (reversed [rising; down entry] hidden word in [restricts] topEKA BALCony) CLAMBAKE |
M |
| 23 | Regularly hoard school dinner (6)
BORSCH ( Russian and Polish beetroot soup that could serve as a dinner) OR (letters 2 and 4 [regularly] of bOaRd + SCH (school) B OR SCH |
B |
| 24 | Defy authority on opening of Bakerloo Line (5)
REBEL (defy authority) RE (concerning; on) + BL (initial letters of [opening of] each of Bakerloo and Line) RE B E L |
E |
| 25 | Oracle of Canterbury? (4)
SEER (an oracle) SEE (the diocese [see] of Canterbury] SEER |
R |
| 27 | Work for ever unfortunately (5)
SERVE (work for) Anagram of (unfortunately) EVER S ERVE* |
S |
Thanks Turnstone for a neat trick and duncanshiell for the great blog with details on the extra words
After “31” themed debut in March, Turnstone returns with the theme of Fibonacci Sequence Numbers.
When half done, I had six of the extra words: spirals, pineapple, sunflower, bees, rabbits, artichokes.
Also had these omitted letters:
F I ? O ? N ? C C I S ? ? U ? ? ? C E ? U M ? E ? ? (26 because I had only 6 words).
Googling the words led to: FIBONACCI SEQUENCE OF NUMBERS
With that, was able to complete the rest of the puzzle. 9d had to be ZERO.
Noticed that the extra words were in the clues that are part of the series …1, 1, 2, 3, 5, 8, 13 and 21.
Thanks again.
I solved quite a few clues when I first looked at this puzzle, but then I got distracted by other crosswords for a while but when I came back to this puzzle that I could see FIBONACCI could guess that clues with extra words were those with Fibonacci numbers and that the extra words were related to Fibonacci or his sequence. I can see how the message suggests which clues to remove words from and which words need to be removed, but I still don’t feel it suggests ‘how the remaining clues must have a word removed’.
I like having a message spelled out by letters omitted from the wordplay better than wordplay generating an extra letter because the answers have to contain certain letters so the grid seems less arbitrarily filled.
Thanks, duncanshiell and Turnstone.
As someone who has quite a few “recreational mathematics” books on the shelf, I thoroughly enjoyed this. Having seen a few appearances in recent years of ‘prime’ in clues to indicate letter positions, when can we expect to see ‘fibonacci’ or similar as a letter position indicator?
Thanks duncanshiell, Turnstone and illipu @1 for pointing out the connection between the extra word clues and the sequence which I’d missed.
No criticism of this crossword, which I enjoyed with it’s multiple references to the Fibonacci series, but the device of adding (or removing) a letter (or word) to (or from) clues seems to have been used an awful lot in the Genius series in recent months. Just an observation, but it will be interesting to see what we get this month.
Many thanks to Turnstone and Duncan.
I liked this a lot. I’m very un-mathematical and I still managed to get the theme 😁 the connections with the extra words were very educational!
Many thanks to Turnstone and Duncan
That was fun. I particularly like the use of 1 down and 1 across to be able to properly do the ‘1, 1’.
Of course, no mention of the FS would be complete without a link to Brian Bilston’s lovely poem: https://brianbilston.com/tag/fibonacci-sequence/
Thanks, Turnstone and Duncan!
This is the first Genius I have completed for a long time, the other one being so long ago I could be imagining it.
As usual with the missing/extra letters it took me too long to complete the phrase, but I got there this time with a few of them left, and enjoyed the penny drop moment. I find it helps a lot to know which letter is special and needed this to finish the last few. I recognised the examples of the sequence from far back in my memory so I didn’t need help finding the Fibonacci clues, and noticing the significance of their numbers was almost the last thing I did. A nice bonus double PDM for me.
I’ve just had a quick scan through the wikipedia page and it really is incredible how such a simple sequence leads to so much maths and examples in nature.
Totally misled by the spell-out Fibonacci sequence of numbers. Where are the letters for of? I used the logical Fibonacci sequence members so got nowhere.
First genius solve, perhaps because of the many anagrams and lack of British-isms! Embarrassingly, it took nearly the whole crossword before I deduced the theme, despite being a math enthusiast. Favorite clue is “Sauce in dish regularly generates bodily chemicals”. Interestingly the wordplay could also be sAuCe In DiSh = ACIDS, so the clue, without enumeration, could have multiple answers.
Thanks Turnstone and Duncan.
I loved this one. Despite having been a maths teacher for 33 years, and the Fibonacci sequence being one of my favourite topics, it took a while for the penny to drop. Once it did, I was off and running. I must admit that it wasn’t until reading this thread that I noticed the eight clues were consecutive terms of the Fibonacci sequence.