## Monthly Prize Puzzle 107 – Review

## April 2021

## A Puzzle by Alchemi

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Congratulations to KiwiColin for winning this month’s prize competition. Colin took advantage of that part of the rules that states “… except that mistakes can be corrected by using the box below and clearly requesting that the earlier entry be withdrawn”.

The instructions were as follows:

*What is the largest number anywhere in the grid? *

We have GOOGOL in 6d which is the 100^{th} power of 10, that is, 10^{100}. In decimal notation, it is written as the digit 1 followed by one hundred zeroes: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

But arguably the biggest named number is googolplex which is 1 followed by 1 googol of zeroes, that is, 10^{googol}. However, we can just keep adding +1 to everything, which is where the aleph numbers come in. When we go beyond any biggest named number, adding +1 to every natural number, we get an infinite set. This is the aleph-null set containing the aleph-null numbers. The first number is the smallest cardinal infinity. Since aleph-null is a cardinal number and every cardinal number plus another cardinal number is still a cardinal number, aleph-null plus aleph-null is still aleph-null. If we want to go beyond this infinite set of cardinal numbers, we need to enter the realm of ordinal numbers. This infinite set is called aleph-one and the numbers in that set are the aleph-one numbers.

And one can spot the second-line Nina that reads ALEPH-ONE which is, in fact, the largest number anywhere in this grid. Hence, ALEPH-ONE is the correct answer.

**Across **

1a Exercise 8 hours before a meal (4,3)

HIGH TEA: An anagram (exercise) of EIGHT (8) and H (hours) followed by (before) A (a)

5a Mayall possibly drags round Mongolian cash (7)

TUGRIKS: The comedian RIK (Mayall possibly) with TUGS (drags) covering (round)

9a A number of Indians challenge results of rapidly electing leaders (5)

CRORE: First letters (leaders) of Challenge, Results, Of, Rapidly and Electing

10a Composer of “Viper” OK after injury (9)

PROKOFIEV: An anagram (after injury) of OF, VIPER and OK

11a Number lists record furniture (3,6)

LOG TABLES: A charade of LOG (record) and TABLES (furniture)

12a Tear-jerker available with newspaper back-number (5)

ONION: A charade of ON (available), I (newspaper) and ON (back-number)

13a Number in church dress can go astray (6,4)

SACRED SONG: An anagram (astray) of DRESS, CAN and GO

16a Suspend time in state (4)

STAY: T (time) contained in (in) SAY (state)

18a Companies keeping black swans (4)

COBS: COS (companies) containing (keeping) B (black)

19a A hat is seen ruining American number (10)

ANESTHESIA: An anagram (ruining) of A, HAT, IS and SEEN

22a Short letter seems incomplete (5)

TERSE: Part of or hidden inside (incomplete) letTER SEems

23a Twist at the beginning of dance number (9)

OCTILLION: Interchange of the first two letters (twist at the beginning) of COTILLION (dance)

26a Specialist oil’s got to work (9)

OTOLOGIST: An anagram (work) of OILS, GOT and TO

27a Feeling even worse with fridge neither cold nor hot (5)

ILLER: CHILLER (fridge) minus C and H (neither cold nor hot)

28a Queen holds soprano’s note, showing virtuosity (7)

MASTERY: MARY (queen) contains (holds) S (soprano’s) and TE (note)

29a Will mentioned one way to discipline students apparently sheltering (7)

LEGATEE: Plausibly through exams of GATE (way to discipline students apparently) going inside a LEE (sheltering)

**Down **

1d Head of restraint demoted, raising these? (7)

HACKLES: The first letter (head) of SHACKLE (restraint) is placed at the last (demoted)

2d German piece of wood good in Scandinavian drink (5)

GLOGG: A charade of G (German) LOG (piece of wood) and G (good)

3d More than 12 support horse starting Epsom Derby (8)

TEENAGED: A charade of TEE (support), NAG (horse) along with first letters (starting) E (Epsom) and D (Derby)

4d Northerner gets up after a shock (5)

APPAL: A reversal as in a downward clue (gets up) of LAPP (northerner) after A (a)

5d Number seen once you are on the beach (9)

THOUSANDS: The old English THOU (once you) is placed upon (are on) SANDS (the beach)

6d Number search engine reported, roughly speaking (6)

GOOGOL: Homophone (reported, roughly speaking) of GOOGLE (search engine)

7d People admitted starts (9)

INITIATES: A double definition of people inducted into a secret or exclusive society (people admitted) and a synonym of the remaining word (starts)

8d Disastrous event – you protect a number (7)

SEVENTY: The words disastrouS EVENT You hide (protect)

14d Numbers of birds taking taxi (4,5)

CUBE ROOTS: COOTS (birds) containing (taking) UBER (taxi)

15d Ordering it on rye is showing your age (9)

SENIORITY: An anagram (ordering) of IT, ON, RYE and IS

17d Giving birth with a couple of scoops? (8)

WHELPING: A charade of W (with) and HELPING or serving of a portion of food (a couple of scoops)

18d Number right to stop in combination with caterpillar (7)

CUTWORM: TWO (number) R (right) to go inside (to stop in) CUM (combination)

20d Fences here, one in oak? (7)

AINTREE: A charade of A (one) IN (in) TREE (oak)

21d Mostly understand Alchemi taking number in sequence (6)

GENOME: GEt (mostly understand) and ME (Alchemi) containing (taking) NO (number)

24d Number collected not ending excessively high (5)

TOTAL: Without the last letters (not ending) of TOo TALl (excessively high)

25d Independent group holding large key (5)

ISLET: A charade of I (independent) and SET (group) containing (holding) L (large)

Thanks to Alchemi and Mrs & Mr BD for playing their parts in this month’s competition process

Well done, Colin, and many thanks to Rahmat for the review. I did get the correct answer but have to confess that I couldn’t fully parse 18d – never thought of ‘cum’ meaning ‘in combination with’ although it does, of course, make perfect sense.

Thanks to Alchemi for the numbers game and to Mr & Mrs BD for organising the MPP.

Well that is a pleasant surprise.

After completing the solve I spent ages staring at the grid and googling things like GOOGOL and eventually decided that the answer must be a bit sneaky and that all the answers were ‘words’ and the largest actual ‘number’ in the grid was 29 (as a clue number) and this is what I submitted. I still wasn’t totally happy with that, so much later, searched the grid again looking for a possible Nina. Thought that ‘Ale Phone’, as well as being a possible means of ordering one’s booze in a contact-free manner, might be worth entering into a Google search. Voila! All became clear.

Thanks to all involved and especially to Rahmat Ali for another of his fine reviews.

Can something that is uncountable and infinite be considered “the largest number”..? I went for G which is Graham’s number, my uncertainty being whether G Log G might be bigger than G, but that depends on the base of the logarithm, which is not defined! Anyway, the editor’s decision is final and it was a great thought-provoking puzzle, the intrinsic elements of which will no doubt come in very useful sometime, somewhere…

Thanks, Alchemi (are you sure that shouldn’t be spelt with a ‘y’?), and also to Rahmat Ali for a fine review.

P.S. Congratulations to KiwiColin! And by the way, in case you ever need to know this for a pub quiz or whatever, Graham’s number may be too large to comprehend but it ends with 7.

Indeed it does

Can you prove that 0 is a number?

Of course – it is the number of correct answers I have submitted to this months MPP…

Thanks to Rahmat Ali for the review and the ‘explanation’ of the answer (although the paragraph explaining aleph-one which I’ve never heard of might as well have been written in Serbo-Croat as far as I’m concerned).

I submitted my answer of Googol before I noticed BD’s bold hint. After seeing that I looked for a more cryptic answer and wondered (like KiwiColin) whether the expected answer might be 29 or even Anesthesia (at 10 characters the largest ‘number’ in the grid) but neither seemed very likely.

Many thanks to Alchemi for the puzzle and to Mr and Mrs BD and congratulations to KiwiColin on his well deserved victory.

Well done, Kiwi Colin.

I thought it would be anesthesia as described by Gazza above.

Thanks to Alchemi.

Congratulations to Kiwi Colin. After a lot of deliberation and explanations on Google we did send in the correct answer – although we were still unsure as to whether it should be googol or 29. Thanks Alchemi for the puzzle and head- scratching involved. Thank you also Mr & Mrs BD.

Well done Kiwi Colin, and thanks Alchemi for a great puzzle and Rahmat for the review. I entered Googol, not spotting the Nina… just when my maths background finally had some chance of being useful, I flunked it, d’oh

Thank you Jane, KiwiColin, Spartacus, Gazza, Fez and others for your words of encouragement on the review. Thanks once again to Alchemi for the splendid creation and to Mrs & Mr BD for hosting the event.

BD should publish the list of answers methinks.

I gave TREE (3) as an answer from Aintree as it referred to the three letter Oak.

Read all about it. Fascinating stuff.

Congratulations to Colin and thanks to Rahmat Ali for the great review.

Thanks again to Alchemi for the fun.

Thank you jean-luc cheval for your words of encouragement on the review.