Amusements-in-Mathematics-34

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was	 asked	 how	 he	would	 intersect	 the	 pen	with	 three	 straight	 fences	 so	 as	 to
enclose	every	pig	in	a	separate	sty.	In	other	words,	all	you	have	to	do	is	to	take
your	pencil	and,	with	 three	straight	strokes	across	 the	square,	enclose	each	pig
separately.	Nothing	could	be	simpler.
The	Irishman	complained	that	the	pigs	would	not	keep	still	while	he	was	putting
up	the	fences.	He	said	that	they	would	all	flock	together,	or	one	obstinate	beast
would	go	into	a	corner	and	flock	all	by	himself.	It	was	pointed	out	to	him	that	for
the	purposes	of	the	puzzle	the	pigs	were	stationary.	He	answered	that	Irish	pigs
are	not	stationery—they	are	pork.	Being	persuaded	to	make	the	attempt,	he	drew
three	lines,	one	of	which	cut	through	a	pig.	When	it	was	explained	that	this	is	not
allowed,	he	protested	that	a	pig	was	no	use	until	you	cut	its	throat.	"Begorra,	if
it's	bacon	ye	want	without	cutting	your	pig,	it	will	be	all	gammon."	We	will	not
do	 the	 Irishman	 the	 injustice	 of	 suggesting	 that	 the	 miserable	 pun	 was
intentional.	However,	he	failed	to	solve	the	puzzle.	Can	you	do	it?
166.—THE	LANDOWNER'S	FENCES.
The	 landowner	 in	 the	 illustration	 is	 consulting	 with	 his	 bailiff	 over	 a	 rather
puzzling	little	question.	He	has	a	large	plan	of	one	of	his	fields,	in	which	there
are	eleven	trees.	Now,	he	wants	to	divide	the	field	into	just	eleven	enclosures	by
means	 of	 straight	 fences,	 so	 that	 every	 enclosure	 shall	 contain	 one	 tree	 as	 a
shelter	 for	 his	 cattle.	How	 is	 he	 to	 do	 it	with	 as	 few	 fences	 as	 possible?	Take
your	pencil	and	draw	straight	lines	across	the	field	until	you	have	marked	off	the
eleven	enclosures	(and	no	more),	and	then	see	how	many	fences	you	require.	Of
course	the	fences	may	cross	one	another.
167.—THE	WIZARD'S	CATS.
A	wizard	placed	ten	cats	inside	a	magic	circle	as	shown	in	our	illustration,	and
hypnotized	 them	so	 that	 they	 should	 remain	 stationary	during	his	pleasure.	He
then	 proposed	 to	 draw	 three	 circles	 inside	 the	 large	 one,	 so	 that	 no	 cat	 could
approach	 another	 cat	 without	 crossing	 a	 magic	 circle.	 Try	 to	 draw	 the	 three
circles	 so	 that	 every	 cat	 has	 its	 own	 enclosure	 and	 cannot	 reach	 another	 cat
without	crossing	a	line.
168.—THE	CHRISTMAS	PUDDING.
"Speaking	of	Christmas	puddings,"	said	the	host,	as	he	glanced	at	the	imposing
delicacy	at	 the	other	 end	of	 the	 table.	 "I	 am	 reminded	of	 the	 fact	 that	 a	 friend
gave	me	a	new	puzzle	the	other	day	respecting	one.	Here	it	is,"	he	added,	diving
into	his	breast	pocket.
"'Problem:	To	find	the	contents,'	I	suppose,"	said	the	Eton	boy.
"No;	the	proof	of	that	is	in	the	eating.	I	will	read	you	the	conditions."
"'Cut	 the	 pudding	 into	 two	 parts,	 each	 of	 exactly	 the	 same	 size	 and	 shape,
without	touching	any	of	the	plums.	The	pudding	is	to	be	regarded	as	a	flat	disc,
not	as	a	sphere.'"
"Why	should	you	 regard	a	Christmas	pudding	as	 a	disc?	And	why	 should	any
reasonable	 person	 ever	 wish	 to	 make	 such	 an	 accurate	 division?"	 asked	 the
cynic.
"It	 is	 just	 a	 puzzle—a	 problem	 in	 dissection."	 All	 in	 turn	 had	 a	 look	 at	 the
puzzle,	but	nobody	succeeded	in	solving	it.	 It	 is	a	 little	difficult	unless	you	are
acquainted	with	the	principle	involved	in	the	making	of	such	puddings,	but	easy
enough	when	you	know	how	it	is	done.
169.—A	TANGRAM	PARADOX.
Many	pastimes	of	great	antiquity,	such	as	chess,	have	so	developed	and	changed
down	the	centuries	that	their	original	inventors	would	scarcely	recognize	them.
This	is	not	the	case	with	Tangrams,	a	recreation	that	appears	to	be	at	least	four
thousand	 years	 old,	 that	 has	 apparently	 never	 been	 dormant,	 and	 that	 has	 not
been	altered	or	"improved	upon"	since	the	legendary	Chinaman	Tan	first	cut	out
the	seven	pieces	shown	in	Diagram	I.	If	you	mark	the	point	B,	midway	between
A	and	C,	on	one	side	of	a	square	of	any	size,	and	D,	midway	between	C	and	E,
on	 an	 adjoining	 side,	 the	 direction	 of	 the	 cuts	 is	 too	 obvious	 to	 need	 further
explanation.	 Every	 design	 in	 this	 article	 is	 built	 up	 from	 the	 seven	 pieces	 of
blackened	 cardboard.	 It	 will	 at	 once	 be	 understood	 that	 the	 possible
combinations	are	infinite.
The	 late	 Mr.	 Sam	 Loyd,	 of	 New	 York,	 who	 published	 a	 small	 book	 of	 very
ingenious	 designs,	 possessed	 the	 manuscripts	 of	 the	 late	 Mr.	 Challenor,	 who
made	a	long	and	close	study	of	Tangrams.	This	gentleman,	it	is	said,	records	that
there	were	originally	seven	books	of	Tangrams,	compiled	in	China	two	thousand
years	 before	 the	Christian	 era.	 These	 books	 are	 so	 rare	 that,	 after	 forty	 years'
residence	in	the	country,	he	only	succeeded	in	seeing	perfect	copies	of	the	first
and	seventh	volumes	with	fragments	of	the	second.	Portions	of	one	of	the	books,
printed	in	gold	leaf	upon	parchment,	were	found	in	Peking	by	an	English	soldier
and	sold	for	three	hundred	pounds.
A	few	years	ago	a	 little	book	came	into	my	possession,	from	the	 library	of	 the
late	 Lewis	 Carroll,	 entitled	The	 Fashionable	 Chinese	 Puzzle.	 It	 contains	 three
hundred	 and	 twenty-three	 Tangram	 designs,	 mostly	 nondescript	 geometrical
figures,	to	be	constructed	from	the	seven	pieces.	It	was	"Published	by	J.	and	E.
Wallis,	42	Skinner	Street,	and	J.	Wallis,	Jun.,	Marine	Library,	Sidmouth"	(South
Devon).	There	 is	 no	 date,	 but	 the	 following	 note	 fixes	 the	 time	of	 publication
pretty	 closely:	 "This	 ingenious	 contrivance	 has	 for	 some	 time	 past	 been	 the
favourite	 amusement	 of	 the	 ex-Emperor	 Napoleon,	 who,	 being	 now	 in	 a
debilitated	 state	 and	 living	 very	 retired,	 passes	 many	 hours	 a	 day	 in	 thus
exercising	 his	 patience	 and	 ingenuity."	 The	 reader	 will	 find,	 as	 did	 the	 great
exile,	 that	 much	 amusement,	 not	 wholly	 uninstructive,	 may	 be	 derived	 from
forming	 the	 designs	 of	 others.	 He	 will	 find	 many	 of	 the	 illustrations	 to	 this
article	quite	easy	to	build	up,	and	some	rather	difficult.	Every	picture	may	thus
be	regarded	as	a	puzzle.
But	 it	 is	 another	 pastime	 altogether	 to	 create	 new	 and	 original	 designs	 of	 a
pictorial	 character,	 and	 it	 is	 surprising	what	 extraordinary	 scope	 the	Tangrams
afford	for	producing	pictures	of	real	life—angular	and	often	grotesque,	it	is	true,
but	 full	 of	 character.	 I	 give	 an	 example	 of	 a	 recumbent	 figure	 (2)	 that	 is
particularly	graceful,	and	only	needs	some	slight	reduction	of	its	angularities	to
produce	an	entirely	satisfactory	outline.
As	I	have	referred	to	the	author	of	Alice	in	Wonderland,	I	give	also	my	designs
of	the	March	Hare	(3)	and	the	Hatter	(4).	I	also	give	an	attempt	at	Napoleon	(5),
and	a	very	excellent	Red	Indian	with	his	Squaw	by	Mr.	Loyd	(6	and	7).	A	large
number	 of	 other	 designs	 will	 be	 found	 in	 an	 article	 by	 me	 in	 The	 Strand