- The Guardian Sat urday 31 Aug ust 2019
62Puzzles
Ye ste r d ay ’s
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SolutionsSandwich sudoku
Medium
Place the digits
from 1-9 in each
row, column and
3x3 block.
The clues outside
the grid show
the sum of the
numbers placed
between the 1 and
9 in that row or
column.
Quick crossword no 15,387
1234 5678910111213 14 151617 181920 2122 23Across
1 Patellae (8)
5 Snoopy’s imprecation (4)
9 Glandular organ in the abdomen (5)
10 Prove superior (7)
11 Without warning (3,2,1,6)
13 Attribute (6)
14 Save from peril (6)
17 Called to remembrance (12)
20 Fashionable — served with ice
cream (1,2,4)
21 Lay to rest (5)
22 Long protruding tooth (4)
23 Soaked by rain (8)
Down
1 Hardy cabbage (4)
2 Surround completely (7)
3 Maize eaten as a vegetable (4,2,3,3)
4 Tree — London district (6)
6 Prize (5)
7 Small, sharp sliver (8)
8 Money paid for work or a service
(12)
12 Flying machine (8)
15 Fastest land animal (7)
16 Peeping Tom (6)
18 Resources (5)
19 Poke (4)Solution no 15,386
GA S TRONOMY
DULAAA
ELEGIAC RANCH
CSP I CSO
LAST SNOOT I ER
UDGSOO
TESTER DI RNDL
TPVCS O
E P I LOGUE AGOG
RNTRWR I
SWAM I DE A L E R S
COL I ET
CHANCEL LORPyrgic: 1 The popularity of e.g.
American Sci-Fi with its “ alternate
universes ” (sic) has no doubt
contributed to the loss of the fi ne
distinction between alternate =
every other (as in: He works alternate
Mondays ) and what used to be
expressed by alternative = another
possibility. The etymological history
and meanings of both words have
been confusingly close.
2 6 more rolled up. He shared out
121 − 2 = 119 = 7 × 19. This must be
the number each received × the new
number in the posse. The new number
can’t be 1, 7 or 119 from the conditions
given; so the number must be 19; the
number of sweets received, 7.
3 From last week we know that the
sum to infi nity of S = 1 + 3x + 5x 2 + 7x 3
+ 9x 4... + ... (2n − 1)x (n − 1)... ] = (x + 1)/(1
− x) 2. This was derived without regardto the sign of x; so Candy replaced x
with −x, giving S* = (1 − x)/(1 + x) 2.
4 Let x in (x, y) represent the
probability of her being at O, and y,
of her being at B. She starts with (1,
0) as she must be at 0; then her state
is (0, 1) (she must be at B!), next her
probable positions are summarised by
(1/2, 2/3) as she has 1/3 of a chance of
being again at 0 and 2/3 of staying at
B. We begin to see a pattern; labelling
(0, 1) the 1st of the sequence the nth
state bracket has integers in it divided
by 3 (n − 1). Now look at the sequence
without the power of 3 dividing it (we
can always reinstate it later): (0, 1),
(1, 2), (2, 7), (7, 20), (20, 7)... given the
nth term we can easily construct the
(n + 1)th by making the fi rst element
of the (n + 1)th the second element of
the (n + 1)th and ensuring the second
number of the (n + 1)th bracket sumswith the fi rst to make 3 (n − 1). In fact
the sequence of the fi rst elements:
0, 1, 2, 7, 20 ... is such that the sum
of any two successive terms equals
a power of 3. [ Point to Ponder : Can
you fi nd a formula for the nth term?]
In fact the nth plus (n + 1)th gives 3 n;
the sequence of the second elements
is the same sequence staggered by 1.
The probabilities are given by [(1,
0),] (0, 1), (1,2)/3, (2, 7)/3 2, (7, 20)/3 3 ,
(20/61)/3 4 ... [Readers familiar with
matrices may like to fi nd the matrix
that transforms the nth state bracket
into the (n + 1)th].
Wordplay: Wordpool c) d) b); EPU
VIGILANTE; Tom Swifty COYLY;
Cryptic TARNISH, IRON, JAR;
Missing Links a) men/tor/mentors;
b) sun/dry/run; c) four/square/
meals; d) a/broad/bean; e) panning/
shot/tower; f ) alter/nation/state.RELEASED BY "What's News" vk.com/wsnws TELEGRAM: t.me/whatsnws