18 Friday March 18 2022 | the timesMindGames
Fill the grid
using the
numbers 1 to 9
only. The
numbers in each
horizontal or
vertical run of
white squares
add up to the
total in the
triangle to its left
or above it. The
same number
may occur more
than once in a
row or column,
but not within
the same run of
white squares.© PUZZLER MEDIAKakuro No 3134
© 2010 KENKEN PUZZLE & TM NEXTOY. DIST. BY UFS, INC. WWW.KENKEN.COMFill the blank squares so that every row and column contains
each of the numbers 1 to 5 once only. The symbols between
the squares indicate whether a number is larger (>) or smaller
(<) than the number next to it.All the digits 1 to 6 must appear in every row and column. In
each thick-line “block”, the target number in the top left-hand
corner is calculated from the digits in all the cells in the block,
using the operation indicated by the symbol.KenKen Difficult No 5531 Futoshiki No 4175
17 16 30 33 23 16 36
23 16 17
17 4
24 36
71428
364
4
34
33717 15
33
7 13
4 10 14
27
3
4 1634
23
3 19
16 14 17
35 2217 17 24RolloutsLet us travel back in time to the pre-
computer era. You are playing in a
chouette, and a huge row breaks
out over how to play Black’s 53 in
this week’s position. As the captain
of the team, you want to play 20/17,
12/7. Other team members want
20/15, 12/9 or 20/12. Because you
are the captain, you insist on your
move and 20/17, 12/7 is played. You
go on to lose the game and face re-
criminations from your team
members. Can you prove that your
choice was the best?
Back in the 1980s the realistic an-
swer was “no”. The only thing you
could do was to take the position
and do a manual rollout. This in-
volved taking each of the possible
moves and then rotating White’s
next roll though all 36 possibilities,
playing each game to the end and
then cataloguing the results. Sadly,
a 36-game trial merely provided
statistical noise. The statisticians
among you will know that you
need many hundreds, if not thou-
sands, of trials to get a meaningful
result. The outcome was that there
was no way of accurately determin-ing the best play, except in very
simple positions.
Now fast-forward to 2022. Ex-
treme Gammon (XG) can perform
thousands of rollouts while you
sleep and give you an accurate an-
swer to any position. It uses a so-
phisticated technique known as
variance reduction to enhance its
rollout capability so that you can be
99 percent certain of its answer.
Now back to our position.
XG is very clear that the best
move is 12/7, 8/5. This play is ag-
gressive but very logical. Black
needs to make his 5-pt to win the
game. Ergo, he should slot it imme-
diately and provide direct covers
for both the 5-pt and the 2-pt. If the
slotted checkers are hit, they can be
recirculated.
Thus, not only has XG settled the
original argument, but it has also
taught our protagonists some
modern theory as well. This type of
analysis is why backgammon
theory has made such giant leaps
forward in the last 30 years. The
1980s expert had to do everything
from first principles. The 2020s
expert has the invaluable support
of XG.© PUZZLER MEDIASlide the letters either horizontally or vertically back into the grid to produce a
completed crossword. Letters are allowed to slide over other lettersEvery letter in this crossword-style grid has been substituted for a number from 1
to 26. Each letter of the alphabet appears in the grid at least once. Use the letters
already provided to work out the identity of further letters. Enter letters in the main
grid and the smaller reference grid until all 26 letters of the alphabet have been
accounted for. Proper nouns are excluded. Yesterday’s solution, rightCluelines Stuck on Codeword? To receive 4 random clues call 0901 293 6262 or
text TIMECODE to 64343. Calls cost £1 plus your telephone company’s network
access charge. Texts cost £1 plus your standard network charge. For the full solution
call 0905 757 0142. Calls cost £1 per minute plus your telephone company’s network
access charge. SP: Spoke, 0333 202 3390 (Mon-Fri, 9am-5.30pm).Winners will receive a Collins English Dictionary & Thesaurus
Solve the puzzle and text in the numbers in the three
shaded boxes. Text TIMES followed by a space, then
your three numbers, eg, TIMES 123, plus your name,
address and postcode to 84901 (UK only), by midnight.
Or enter by phone. Call 09012 925274 (ROI 1516 303 501)
by midnight. Leave your three answer numbers (in any
order) and your contact details.
Calls cost £1 (ROI €1.50) plus your telephone company’s
network access charge. Texts cost £1 plus your standard
network charge. Winners will be picked at random from
all correct answers received. One draw per week. Lines
close at midnight tonight. If you call or text after this
time you will not be entered but will still be charged.
SP: Spoke, 0333 202 3390 (Mon-Fri, 9am-5.30pm).Fill the grid so
that every
column, every
row and every
3x2 box contains
the digits 1 to 6What are your favourite puzzles in MindGames?
Email: [email protected]Lay tracks to enable the train to travel from village A to village
B. The numbers indicate how many sections of rail go in each
row and column. There are only straight rails and curved rails.
The track cannot cross itself.Win a Dictionary & Thesaurus
Train Tracks No 1583
Lexica No 6269 No 6270
ATEI GINEOCUR EE LS IL NK GO DME ECOOITRYL YE AM OE RO TRCodeword No 4539
Winning Move
Backgammon Chris Bray
W____W^
árDWDkgW4]
àDpDbDpDW]
ßpDWhpDWD]
ÞDq)PDWDW]
ÝWDWDW)p0]
Ü)N!WDWDW]
ÛWDWHWDP)]
Ú$WDWDRDK]
WÁÂÃÄÅÆÇÈW
White to play.
This position is from Spassky-
Fischer, World Championship
(Game 11) 1972.In the 1972 World Championship
clash Bobby Fischer convincingly
defeated Boris Spassky by the
score of 12½-8½. Spassky’s best
win came in the eleventh game.
Although he is a piece down he
can capture the black rook on h8
or the black knight on d6. In fact
he found something better. Can
you see it?