Amusements-in-Mathematics-49

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250.—THE	GRAND	TOUR.
One	of	the	everyday	puzzles	of	life	is	the	working	out	of	routes.	If	you	are	taking
a	holiday	on	your	bicycle,	or	 a	motor	 tour,	 there	always	arises	 the	question	of
how	 you	 are	 to	 make	 the	 best	 of	 your	 time	 and	 other	 resources.	 You	 have
determined	to	get	as	far	as	some	particular	place,	to	include	visits	to	such-and-
such	a	town,	to	try	to	see	something	of	special	interest	elsewhere,	and	perhaps	to
try	to	look	up	an	old	friend	at	a	spot	that	will	not	take	you	much	out	of	your	way.
Then	you	have	to	plan	your	route	so	as	to	avoid	bad	roads,	uninteresting	country,
and,	if	possible,	the	necessity	of	a	return	by	the	same	way	that	you	went.	With	a
map	before	 you,	 the	 interesting	puzzle	 is	 attacked	 and	 solved.	 I	will	 present	 a
little	poser	based	on	these	lines.
I	 give	 a	 rough	 map	 of	 a	 country—it	 is	 not	 necessary	 to	 say	 what	 particular
country—the	 circles	 representing	 towns	 and	 the	 dotted	 lines	 the	 railways
connecting	 them.	Now	there	 lived	 in	 the	 town	marked	A	a	man	who	was	born
there,	and	during	the	whole	of	his	life	had	never	once	left	his	native	place.	From
his	youth	upwards	he	had	been	very	industrious,	sticking	incessantly	to	his	trade,
and	had	no	desire	whatever	 to	 roam	abroad.	However,	 on	 attaining	his	 fiftieth
birthday	he	decided	to	see	something	of	his	country,	and	especially	to	pay	a	visit
to	a	very	old	friend	living	at	the	town	marked	Z.	What	he	proposed	was	this:	that
he	would	start	from	his	home,	enter	every	town	once	and	only	once,	and	finish
his	journey	at	Z.	As	he	made	up	his	mind	to	perform	this	grand	tour	by	rail	only,
he	found	it	rather	a	puzzle	to	work	out	his	route,	but	he	at	length	succeeded	in
doing	so.	How	did	he	manage	it?	Do	not	forget	that	every	town	has	to	be	visited
once,	and	not	more	than	once.
251.—WATER,	GAS,	AND	ELECTRICITY.
There	 are	 some	 half-dozen	 puzzles,	 as	 old	 as	 the	 hills,	 that	 are	 perpetually
cropping	up,	and	there	is	hardly	a	month	in	the	year	that	does	not	bring	inquiries
as	 to	 their	 solution.	 Occasionally	 one	 of	 these,	 that	 one	 had	 thought	 was	 an
extinct	volcano,	bursts	 into	eruption	in	a	surprising	manner.	I	have	received	an
extraordinary	number	of	 letters	 respecting	 the	ancient	puzzle	 that	 I	have	called
"Water,	Gas,	and	Electricity."	It	is	much	older	than	electric	lighting,	or	even	gas,
but	 the	new	dress	brings	 it	 up	 to	date.	The	puzzle	 is	 to	 lay	on	water,	 gas,	 and
electricity,	from	W,	G,	and	E,	to	each	of	the	three	houses,	A,	B,	and	C,	without
any	 pipe	 crossing	 another.	 Take	 your	 pencil	 and	 draw	 lines	 showing	 how	 this
should	be	done.	You	will	soon	find	yourself	landed	in	difficulties.
252.—A	PUZZLE	FOR	MOTORISTS.
Eight	 motorists	 drove	 to	 church	 one	 morning.	 Their	 respective	 houses	 and
churches,	 together	with	 the	 only	 roads	 available	 (the	 dotted	 lines),	 are	 shown.
One	went	 from	his	 house	A	 to	 his	 church	A,	 another	 from	his	 house	B	 to	 his
church	B,	another	from	C	to	C,	and	so	on,	but	 it	was	afterwards	found	that	no
driver	ever	crossed	the	track	of	another	car.	Take	your	pencil	and	try	to	trace	out
their	various	routes.
253.—A	BANK	HOLIDAY	PUZZLE.
Two	friends	were	spending	their	bank	holiday	on	a	cycling	trip.	Stopping	for	a
rest	 at	 a	 village	 inn,	 they	 consulted	 a	 route	map,	 which	 is	 represented	 in	 our
illustration	 in	 an	 exceedingly	 simplified	 form,	 for	 the	 puzzle	 is	 interesting
enough	without	all	the	original	complexities.	They	started	from	the	town	in	the
top	 left-hand	 corner	marked	A.	 It	will	 be	 seen	 that	 there	 are	 one	 hundred	 and
twenty	 such	 towns,	 all	 connected	 by	 straight	 roads.	Now	 they	 discovered	 that
there	 are	 exactly	 1,365	 different	 routes	 by	 which	 they	 may	 reach	 their
destination,	 always	 travelling	 either	 due	 south	 or	 due	 east.	 The	 puzzle	 is	 to
discover	which	town	is	their	destination.
Of	course,	if	you	find	that	there	are	more	than	1,365	different	routes	to	a	town	it
cannot	be	the	right	one.
254.—THE	MOTOR-CAR	TOUR.
In	the	above	diagram	the	circles	represent	towns	and	the	lines	good	roads.	In	just
how	many	different	ways	can	a	motorist,	starting	from	London	(marked	with	an
L),	make	a	tour	of	all	these	towns,	visiting	every	town	once,	and	only	once,	on	a
tour,	and	always	coming	back	to	London	on	the	last	ride?	The	exact	reverse	of
any	route	is	not	counted	as	different.
255.—THE	LEVEL	PUZZLE.
This	is	a	simple	counting	puzzle.	In	how	many	different	ways	can	you	spell	out
the	word	LEVEL	by	placing	the	point	of	your	pencil	on	an	L	and	then	passing
along	the	lines	from	letter	to	letter.	You	may	go	in	any	direction,	backwards	or
forwards.	Of	 course	you	are	not	 allowed	 to	miss	 letters—that	 is	 to	 say,	 if	 you
come	to	a	letter	you	must	use	it.
256.—THE	DIAMOND	PUZZLE.
IN	 how	 many	 different	 ways	 may	 the	 word	 DIAMOND	 be	 read	 in	 the
arrangement	shown?	You	may	start	wherever	you	like	at	a	D	and	go	up	or	down,
backwards	 or	 forwards,	 in	 and	 out,	 in	 any	 direction	 you	 like,	 so	 long	 as	 you
always	pass	from	one	letter	to	another	that	adjoins	it.	How	many	ways	are	there?
257.—THE	DEIFIED	PUZZLE.
In	how	many	different	ways	may	the	word	DEIFIED	be	read	in	this	arrangement
under	 the	same	conditions	as	 in	 the	 last	puzzle,	with	 the	addition	 that	you	can
use	any	letters	twice	in	the	same	reading?
258.—THE	VOTERS'	PUZZLE.
Here	we	have,	 perhaps,	 the	most	 interesting	 form	of	 the	puzzle.	 In	how	many