# Miscellaneous pure mathematics

Sections: Pure, Games, Geometry, Physics, Large.

If you make any progress on any of these puzzles, or if you'd like to contribute a new one, then please do tell me about it.

Puzzles in this section:

### 13 points on a circle

13 points on a circle

Q: ??? 13 points in the plane are such that given any three some fourth one lies on the same circle as those three. Show that all 13 are cocircular.
[Julian Gilbey, 13 Jun 1996]

### Glass-wall Roman numeral determinant

Glass-wall Roman numeral determinant

Q: Find a determinant identity using Roman numerals that can be painted on a glass wall and be valid from both sides. To be interesting it should involve the Roman decrement convention (eg IV = V - I).
[rec.puzzles]

### Pyewackett cover

Pyewackett cover

Q: ??? The cover of Pyewackett's "This Crazy Paradise" album includes an image something like the following:

Is this significant? If so, why?
[29 Aug 1996]

### Sequences

Sequences [11 Apr 1996]

Q: ??? What is the next term in each of the following?:

• 0, 1, 10, 100, 11, 1000, 3, 20 [Neal Sloane]
• 1, 12, 66, 220, 495, 792 [rec.puzzles, 2 Feb 1996]
• 25, 7, 49, 3, 17, 6, 67, 1, 39, 4, 75 [rec.puzzles, 4 Apr 1996]

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### Sir Arthur Eddington's Cricket Problem

Sir Arthur Eddington's Cricket Problem

Extract from the score of a cricket match between Eastershire and Westershire:

EASTERSHIRE - Second Innings:

 Total A. A. Atkins 6 Bodkins 8 D. D. Dawkins 6 Hawkins 6 Jenkins (J) 5 Larkins 4 Meakins 7 Hon. P. P. Perkins 11 Capt. S. S. Simkins 6 Tomkins 0 Wilkins 1 Extras 0 60

Bowling Analysis:

 Overs Maidens Runs Wickets Pitchwell 12.1 2 14 8 Speedwell 6 0 15 1 Tosswell 7 5 31 1

The score was entirely in singles and fours, with no catches, no-balls, or short runs. Speedwell and Tosswell each had only one spell of bowling. Pitchwell bowled the first over, with Mr Atkins taking the first ball. Speedwell was the other opening bowler.

Q:

1. Whose wickets were taken by Speedwell and Tosswell?
2. Who was not out?
3. What was the score at the fall of each wicket?

[Editor's note: This is a brilliant and complex puzzle. The spoiler has two complete solutions, but each is well over 1000 words. I do urge you to have a serious crack at it before you look at the spoiler. Here are some hints to get you started if you want, hidden by rot13 so as not to spoil the fun:

1. Jub objyrq rnpu bire?
2. Jung fpbevat fgebxrf jrer gurer va Gbffjryy'f aba-znvqra biref? Jub pbhyq unir fpberq gurz?
3. Jung qbrf guvf gryy lbh nobhg Obqxvaf?
4. Jung pna lbh fnl nobhg jung unccrarq va gur svefg srj biref?]

### Triple minor sets

Triple minor sets

Q: In English country dancing a "triple minor" set consists of a line of n couples, numbered 1, 2, 3, 1, 2, 3, ... from the top: each block of three couples 1, 2, 3 is called a "minor set". At each turn of the dance the 1s move down one place and the 2s move up one place, and the 2s and 3s then swap numbers to form new minor sets numbered 1, 2, 3 (omitting the new top couple). During any turn only those in complete minor sets dance (experienced dancers sometimes do a fudged version in an incomplete minor set, but ignore this for now).

The 1s move down the set: when they reach the bottom - 1 position they immediately move to the bottom, and wait two turns to restart as 3s. The others move (more slowly) up the set, alternately dancing as 2s and 3s: when they reach the top - 1 position they wait one turn, dance once as 2s to get to top place, and wait two turns to restart as 1s. Normally the dance has t = 3u + 1 turns, for some u.

Many people believe that couples spend a greater proportion of time dancing if n is not a multiple of 3. Is this true?
[David Barnert, ECD discussion list, 19 Jun 2001]

Q: Generalise to minor sets of m couples, as follows: after the first turn the 1s have moved down one place and as before become the top of new minor sets, so the couple below them change their number to 2, the next couple to 3, etc. End effects in this case are left as an exercise for the reader. Let n = pm + q, and while we're at it generalise the number of turns to t = um + v. How does the number of couples affect the efficiency?

Notes: There are many "duple minor" English country dances, with m = 2. There are quite a few "triple minors", with m = 3 as above, which were particularly popular in the 17th and 18th centuries because they tend to give the 2s and 3s more flirting time. Many Scottish country dances also have this type of progression, almost always with n = 4, m = 3 and t = 8 (although with such a small n the two sets of end effects meet...). Hampstead Manor is a "quadruple minor", with m = 4, but I don't know details, or any other such dances. A dance with m = 3 or 4 should not be confused with a duple minor that has a triple or quadruple progression. Nor is it the same as a double contra, such as Jim Kitch's Double Chocolate, which is different again.

### Two black neighbours implies black

Two black neighbours implies black

Q: Consider a 10 x 10 array of white squares. Colour some initial set black. Now, if a square has two black orthogonal neighbours, colour it black. Repeat as much as possible. Show that if the initial set has 9 squares we cannot end with the whole array black.
[Dima Fon-Der-Flaass, 24 Feb 1997]

Q: More generally, find the minimum initial # blacks required to turn an m x n array black.

Sections: Pure, Games, Geometry, Physics, Large.

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