Amusements-in-Mathematics-104

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to	the	desired	solutions.
89.—ADDING	THE	DIGITS.—solution
The	smallest	possible	sum	of	money	is	£1,	8s.	9¾d.,	the	digits	of	which	add	to
25.
90.—THE	CENTURY	PUZZLE.—solution
The	problem	of	expressing	the	number	100	as	a	mixed	number	or	fraction,	using
all	 the	 nine	 digits	 once,	 and	 once	 only,	 has,	 like	 all	 these	 digital	 puzzles,	 a
fascinating	side	to	it.	The	merest	tyro	can	by	patient	trial	obtain	correct	results,
and	 there	 is	 a	 singular	 pleasure	 in	 discovering	 and	 recording	 each	 new
arrangement	 akin	 to	 the	 delight	 of	 the	 botanist	 in	 finding	 some	 long-sought
plant.	It	is	simply	a	matter	of	arranging	those	nine	figures	correctly,	and	yet	with
the	thousands	of	possible	combinations	that	confront	us	the	task	is	not	so	easy	as
might	at	first	appear,	if	we	are	to	get	a	considerable	number	of	results.	Here	are
eleven	answers,	including	the	one	I	gave	as	a	specimen:—
962148/537
961752/438
961428/357
941578/263
917524/836
915823/647
915742/638
823546/197
817524/396
815643/297
369258/714
Now,	 as	 all	 the	 fractions	 necessarily	 represent	 whole	 numbers,	 it	 will	 be
convenient	 to	 deal	with	 them	 in	 the	 following	 form:	 96	 +	 4,	 94	 +	 6,	 91	 +	 9,
82	+	18,	81	+	19,	and	3	+	97.
With	any	whole	number	the	digital	roots	of	the	fraction	that	brings	it	up	to	100
will	always	be	of	one	particular	form.	Thus,	in	the	case	of	96	+	4,	one	can	say	at
once	that	if	any	answers	are	obtainable,	then	the	roots	of	both	the	numerator	and
the	denominator	of	the	fraction	will	be	6.	Examine	the	first	 three	arrangements
given	above,	and	you	will	find	that	this	is	so.	In	the	case	of	94	+	6	the	roots	of
the	numerator	and	denominator	will	be	respectively	3—2,	in	the	case	of	91	+	9
and	of	82	+	18	they	will	be	9—8,	in	the	case	of	81	+	19	they	will	be	9—9,	and	in
the	case	of	3	+	97	they	will	be	3—3.	Every	fraction	that	can	be	employed	has,
therefore,	its	particular	digital	root	form,	and	you	are	only	wasting	your	time	in
unconsciously	attempting	to	break	through	this	law.
Every	 reader	 will	 have	 perceived	 that	 certain	 whole	 numbers	 are	 evidently
impossible.	Thus,	if	there	is	a	5	in	the	whole	number,	there	will	also	be	a	nought
or	a	second	5	in	the	fraction,	which	are	barred	by	the	conditions.	Then	multiples
of	 10,	 such	 as	 90	 and	 80,	 cannot	 of	 course	 occur,	 nor	 can	 the	whole	 number
conclude	with	a	9,	 like	89	and	79,	because	the	fraction,	equal	to	11	or	21,	will
have	1	in	the	last	place,	and	will	therefore	repeat	a	figure.	Whole	numbers	that
repeat	a	figure,	such	as	88	and	77,	are	also	clearly	useless.	These	cases,	as	I	have
said,	are	all	obvious	to	every	reader.	But	when	I	declare	that	such	combinations
as	98	+	2,	92	+	8,	86	+	14,	83	+	17,	74	+	26,	etc.,	etc.,	are	to	be	at	once	dismissed
as	 impossible,	 the	 reason	 is	 not	 so	 evident,	 and	 I	 unfortunately	 cannot	 spare
space	to	explain	it.
But	 when	 all	 those	 combinations	 have	 been	 struck	 out	 that	 are	 known	 to	 be
impossible,	 it	 does	 not	 follow	 that	 all	 the	 remaining	 "possible	 forms"	 will
actually	work.	The	elemental	form	may	be	right	enough,	but	there	are	other	and
deeper	considerations	that	creep	in	to	defeat	our	attempts.	For	example,	98	+	2	is
an	 impossible	combination,	because	we	are	able	 to	say	at	once	 that	 there	 is	no
possible	form	for	 the	digital	roots	of	 the	fraction	equal	 to	2.	But	 in	 the	case	of
97	+	3	there	is	a	possible	form	for	the	digital	roots	of	the	fraction,	namely,	6—5,
and	it	is	only	on	further	investigation	that	we	are	able	to	determine	that	this	form
cannot	in	practice	be	obtained,	owing	to	curious	considerations.	The	working	is
greatly	simplified	by	a	process	of	elimination,	based	on	such	considerations	as
that	 certain	multiplications	 produce	 a	 repetition	 of	 figures,	 and	 that	 the	whole
number	cannot	be	from	12	to	23	inclusive,	since	in	every	such	case	sufficiently
small	denominators	are	not	available	for	forming	the	fractional	part.
91.—MORE	MIXED	FRACTIONS.—solution
The	point	of	the	present	puzzle	lies	in	the	fact	that	the	numbers	15	and	18	are	not
capable	of	solution.	There	is	no	way	of	determining	this	without	trial.	Here	are
answers	for	the	ten	possible	numbers:—
95472/1368 = 13;
96435/1287 = 14;
123576/894 = 16;
613258/947 = 20;
159432/786 = 27;
249756/813 = 36;
275148/396 = 40;
651892/473 = 69;
593614/278 = 72;
753648/192 = 94.
I	have	only	found	the	one	arrangement	for	each	of	the	numbers	16,	20,	and	27;
but	the	other	numbers	are	all	capable	of	being	solved	in	more	than	one	way.	As
for	 15	 and	 18,	 though	 these	may	 be	 easily	 solved	 as	 a	 simple	 fraction,	 yet	 a
"mixed	fraction"	assumes	the	presence	of	a	whole	number;	and	though	my	own
idea	 for	 dodging	 the	 conditions	 is	 the	 following,	 where	 the	 fraction	 is	 both
complex	and	mixed,	it	will	be	fairer	to	keep	exactly	to	the	form	indicated:—