Amusements-in-Mathematics-89

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"But	they	do	fit,"	said	Uncle	John.	"Try	it,	and	you	will	see."
Later	in	the	evening	Reginald	and	George,	were	seen	in	a	corner	with	their	heads
together,	trying	to	catch	that	elusive	little	square,	and	it	is	only	fair	to	record	that
before	 they	 retired	 for	 the	 night	 they	 succeeded	 in	 securing	 their	 prey,	 though
some	others	of	the	company	failed	to	see	it	when	captured.	Can	the	reader	solve
the	little	mystery?
UNCLASSIFIED	PROBLEMS.
"A	snapper	up	of	unconsidered	trifles."
Winter's	Tale,	iv.	2.
414.—WHO	WAS	FIRST?
Anderson,	Biggs,	and	Carpenter	were	staying	together	at	a	place	by	the	seaside.
One	day	they	went	out	in	a	boat	and	were	a	mile	at	sea	when	a	rifle	was	fired	on
shore	in	their	direction.	Why	or	by	whom	the	shot	was	fired	fortunately	does	not
concern	us,	as	no	information	on	these	points	is	obtainable,	but	from	the	facts	I
picked	up	we	can	get	material	for	a	curious	little	puzzle	for	the	novice.
It	 seems	 that	 Anderson	 only	 heard	 the	 report	 of	 the	 gun,	 Biggs	 only	 saw	 the
smoke,	and	Carpenter	merely	saw	the	bullet	strike	the	water	near	them.	Now,	the
question	arises:	Which	of	them	first	knew	of	the	discharge	of	the	rifle?
415.—A	WONDERFUL	VILLAGE.
There	is	a	certain	village	in	Japan,	situated	in	a	very	low	valley,	and	yet	the	sun
is	nearer	to	the	inhabitants	every	noon,	by	3,000	miles	and	upwards,	than	when
he	either	rises	or	sets	to	these	people.	In	what	part	of	the	country	is	the	village
situated?
416.—A	CALENDAR	PUZZLE.
If	the	end	of	the	world	should	come	on	the	first	day	of	a	new	century,	can	you
say	what	are	the	chances	that	it	will	happen	on	a	Sunday?
417.—THE	TIRING	IRONS.
The	illustration	represents	one	of	the	most	ancient	of	all	mechanical	puzzles.	Its
origin	 is	 unknown.	 Cardan,	 the	 mathematician,	 wrote	 about	 it	 in	 1550,	 and
Wallis	 in	 1693;	 while	 it	 is	 said	 still	 to	 be	 found	 in	 obscure	 English	 villages
(sometimes	deposited	in	strange	places,	such	as	a	church	belfry),	made	of	iron,
and	appropriately	called	"tiring-irons,"	and	to	be	used	by	the	Norwegians	to-day
as	a	lock	for	boxes	and	bags.	In	the	toyshops	it	is	sometimes	called	the	"Chinese
rings,"	 though	 there	 seems	 to	 be	 no	 authority	 for	 the	 description,	 and	 it	more
frequently	goes	by	the	unsatisfactory	name	of	"the	puzzling	rings."	The	French
call	it	"Baguenaudier."
The	puzzle	will	be	seen	to	consist	of	a	simple	loop	of	wire	fixed	in	a	handle	to	be
held	in	the	left	hand,	and	a	certain	number	of	rings	secured	by	wires	which	pass
through	holes	in	the	bar	and	are	kept	there	by	their	blunted	ends.	The	wires	work
freely	 in	 the	bar,	but	cannot	come	apart	 from	it,	nor	can	 the	wires	be	removed
from	the	rings.	The	general	puzzle	is	to	detach	the	loop	completely	from	all	the
rings,	and	then	to	put	them	all	on	again.
Now,	it	will	be	seen	at	a	glance	that	the	first	ring	(to	the	right)	can	be	taken	off	at
any	time	by	sliding	it	over	the	end	and	dropping	it	through	the	loop;	or	it	may	be
put	on	by	reversing	the	operation.	With	this	exception,	the	only	ring	that	can	ever
be	removed	is	the	one	that	happens	to	be	a	contiguous	second	on	the	loop	at	the
right-hand	end.	Thus,	with	all	the	rings	on,	the	second	can	be	dropped	at	once;
with	the	first	ring	down,	you	cannot	drop	the	second,	but	may	remove	the	third;
with	the	first	three	rings	down,	you	cannot	drop	the	fourth,	but	may	remove	the
fifth;	and	so	on.	It	will	be	found	that	the	first	and	second	rings	can	be	dropped
together	or	put	on	together;	but	to	prevent	confusion	we	will	throughout	disallow
this	 exceptional	 double	 move,	 and	 say	 that	 only	 one	 ring	 may	 be	 put	 on	 or
removed	at	a	time.
We	can	thus	take	off	one	ring	in	1	move;	two	rings	in	2	moves;	three	rings	in	5
moves;	 four	 rings	 in	 10	 moves;	 five	 rings	 in	 21	 moves;	 and	 if	 we	 keep	 on
doubling	 (and	 adding	 one	 where	 the	 number	 of	 rings	 is	 odd)	 we	 may	 easily
ascertain	the	number	of	moves	for	completely	removing	any	number	of	rings.	To
get	off	all	the	seven	rings	requires	85	moves.	Let	us	look	at	the	five	moves	made
in	removing	the	first	three	rings,	the	circles	above	the	line	standing	for	rings	on
the	loop	and	those	under	for	rings	off	the	loop.
	AMUSEMENTS IN MATHEMATICS.
	UNCLASSIFIED PROBLEMS.