Amusements-in-Mathematics-22

Prévia do material em texto

113.—THE	TORN	NUMBER.
I	had	the	other	day	in	my	possession	a	label	bearing	the	number	3	0	2	5	in	large
figures.	This	got	accidentally	torn	in	half,	so	that	3	0	was	on	one	piece	and	2	5
on	the	other,	as	shown	on	the	illustration.	On	looking	at	these	pieces	I	began	to
make	a	calculation,	scarcely	conscious	of	what	I	was	doing,	when	I	discovered
this	little	peculiarity.	If	we	add	the	3	and	the	2	5	together	and	square	the	sum	we
get	as	the	result	the	complete	original	number	on	the	label!	Thus,	30	added	to	25
is	55,	and	55	multiplied	by	55	is	3025.	Curious,	is	it	not?	Now,	the	puzzle	is	to
find	 another	 number,	 composed	 of	 four	 figures,	 all	 different,	 which	 may	 be
divided	in	the	middle	and	produce	the	same	result.
114.—CURIOUS	NUMBERS.
The	number	48	has	this	peculiarity,	that	if	you	add	1	to	it	the	result	is	a	square
number	(49,	the	square	of	7),	and	if	you	add	1	to	its	half,	you	also	get	a	square
number	(25,	the	square	of	5).	Now,	there	is	no	limit	to	the	numbers	that	have	this
peculiarity,	 and	 it	 is	 an	 interesting	 puzzle	 to	 find	 three	 more	 of	 them—the
smallest	possible	numbers.	What	are	they?
115.—A	PRINTER'S	ERROR.
In	a	certain	article	a	printer	had	to	set	up	the	figures	54	×	23,	which,	of	course,
means	that	the	fourth	power	of	5	(625)	is	to	be	multiplied	by	the	cube	of	2	(8),
the	product	of	which	 is	5,000.	But	he	printed	54	×	23	 as	5	4	2	3,	which	 is	not
correct.	Can	you	place	four	digits	in	the	manner	shown,	so	that	it	will	be	equally
correct	if	the	printer	sets	it	up	aright	or	makes	the	same	blunder?
116.—THE	CONVERTED	MISER.
Mr.	 Jasper	 Bullyon	 was	 one	 of	 the	 very	 few	 misers	 who	 have	 ever	 been
converted	to	a	sense	of	 their	duty	towards	their	 less	fortunate	fellow-men.	One
eventful	night	he	counted	out	his	accumulated	wealth,	and	resolved	to	distribute
it	amongst	the	deserving	poor.
He	found	that	if	he	gave	away	the	same	number	of	pounds	every	day	in	the	year,
he	could	exactly	spread	it	over	a	twelvemonth	without	there	being	anything	left
over;	 but	 if	 he	 rested	 on	 the	Sundays,	 and	 only	 gave	 away	 a	 fixed	 number	 of
pounds	 every	weekday,	 there	would	be	one	 sovereign	 left	 over	on	New	Year's
Eve.	 Now,	 putting	 it	 at	 the	 lowest	 possible,	 what	 was	 the	 exact	 number	 of
pounds	that	he	had	to	distribute?
Could	any	question	be	simpler?	A	sum	of	pounds	divided	by	one	number	of	days
leaves	no	remainder,	but	divided	by	another	number	of	days	leaves	a	sovereign
over.	That	is	all;	and	yet,	when	you	come	to	tackle	this	little	question,	you	will
be	surprised	that	it	can	become	so	puzzling.
117.—A	FENCE	PROBLEM.
The	practical	usefulness	of	puzzles	is	a	point	that	we	are	liable	to	overlook.	Yet,
as	 a	matter	of	 fact,	 I	 have	 from	 time	 to	 time	 received	quite	 a	 large	number	of
letters	 from	 individuals	 who	 have	 found	 that	 the	 mastering	 of	 some	 little
principle	 upon	 which	 a	 puzzle	 was	 built	 has	 proved	 of	 considerable	 value	 to
them	 in	 a	most	unexpected	way.	 Indeed,	 it	may	be	accepted	as	 a	good	maxim
that	 a	 puzzle	 is	 of	 little	 real	 value	 unless,	 as	 well	 as	 being	 amusing	 and
perplexing,	 it	 conceals	 some	 instructive	 and	 possibly	 useful	 feature.	 It	 is,
however,	very	curious	how	these	little	bits	of	acquired	knowledge	dovetail	into
the	occasional	requirements	of	everyday	life,	and	equally	curious	to	what	strange
and	mysterious	uses	some	of	our	readers	seem	to	apply	them.	What,	for	example,
can	be	the	object	of	Mr.	Wm.	Oxley,	who	writes	to	me	all	the	way	from	Iowa,	in
wishing	 to	 ascertain	 the	 dimensions	 of	 a	 field	 that	 he	 proposes	 to	 enclose,
containing	just	as	many	acres	as	there	shall	be	rails	in	the	fence?
The	man	wishes	 to	fence	 in	a	perfectly	square	field	which	 is	 to	contain	 just	as
many	 acres	 as	 there	 are	 rails	 in	 the	 required	 fence.	Each	 hurdle,	 or	 portion	 of
fence,	is	seven	rails	high,	and	two	lengths	would	extend	one	pole	(16½	ft.):	that
is	to	say,	there	are	fourteen	rails	to	the	pole,	lineal	measure.	Now,	what	must	be
the	size	of	the	field?
118.—CIRCLING	THE	SQUARES.
The	puzzle	is	to	place	a	different	number	in	each	of	the	ten	squares	so	that	the
sum	of	the	squares	of	any	two	adjacent	numbers	shall	be	equal	to	the	sum	of	the
squares	 of	 the	 two	numbers	 diametrically	 opposite	 to	 them.	The	 four	 numbers
placed,	 as	 examples,	must	 stand	 as	 they	 are.	The	 square	of	 16	 is	 256,	 and	 the
square	of	2	is	4.	Add	these	together,	and	the	result	is	260.	Also—the	square	of	14
is	 196,	 and	 the	 square	 of	 8	 is	 64.	 These	 together	 also	 make	 260.	 Now,	 in
precisely	the	same	way,	B	and	C	should	be	equal	to	G	and	H	(the	sum	will	not
necessarily	be	260),	A	and	K	to	F	and	E,	H	and	I	to	C	and	D,	and	so	on,	with	any
two	adjoining	squares	in	the	circle.
All	 you	 have	 to	 do	 is	 to	 fill	 in	 the	 remaining	 six	 numbers.	 Fractions	 are	 not
allowed,	and	I	shall	show	that	no	number	need	contain	more	than	two	figures.
119.—RACKBRANE'S	LITTLE	LOSS.
Professor	Rackbrane	was	spending	an	evening	with	his	old	friends,	Mr.	and	Mrs.
Potts,	and	they	engaged	in	some	game	(he	does	not	say	what	game)	of	cards.	The
professor	lost	the	first	game,	which	resulted	in	doubling	the	money	that	both	Mr.
and	Mrs.	Potts	had	 laid	on	 the	 table.	The	second	game	was	 lost	by	Mrs.	Potts,
which	doubled	the	money	then	held	by	her	husband	and	the	professor.	Curiously
enough,	the	third	game	was	lost	by	Mr.	Potts,	and	had	the	effect	of	doubling	the
money	 then	 held	 by	 his	 wife	 and	 the	 professor.	 It	 was	 then	 found	 that	 each
person	had	exactly	 the	same	money,	but	 the	professor	had	lost	five	shillings	 in
the	course	of	play.	Now,	 the	professor	 asks,	what	was	 the	 sum	of	money	with
which	he	sat	down	at	the	table?	Can	you	tell	him?
120.—THE	FARMER	AND	HIS	SHEEP.
Farmer	Longmore	had	a	curious	aptitude	 for	arithmetic,	and	was	known	 in	his
district	as	the	"mathematical	farmer."	The	new	vicar	was	not	aware	of	this	fact
when,	meeting	his	worthy	parishioner	one	day	in	the	lane,	he	asked	him	in	the