Mathematical & Logical Puzzles -A4

13.     The Tribulations Of A Father Of Five

Let A,B,C,D,E be the five children in descending order of age.

Left Bank	Crossing	Right Bank
ABCDE	-	-
ACE	BD ®	-
ACE	¬ B	D
BE	AC ®	D
BE	¬ AD	C
BD	AE ®	C
BD	¬ CE	A
BD	CE ®	A
BD	¬ -	ACE
-	BD ®	ACE
-	-	ABCDE

The minimum number of crossings is nine with three crossings per child.  Note that if C and E have only one crossing each, the total number of crossings is reduced from nine to seven.

14.     Three Wives & Their Jealous Husbands

Let the three husbands be A,B and C and their wives a,b and c respectively.  The following moves are typical of a minimum solution with men rowing where possible.

Left Bank	Crossing	Right Bank
ABCabc	-	-
ACac	Bb ®	-
ACac	¬ B	b
ABC	ac ®	b
ABC	¬ a	bc
Aa	BC ®	bc
Aa	¬ Bb	Cc
ab	AB ®	Cc
ab	¬ c	ABC
b	ac ®	ABC
b	¬ B	ACac
-	Bb®	ACac
-	-	ABCabc

Note that only a, B, and c need be able to row.

15.     Four Wives & Their Jealous Husbands

Let the husbands be ABCD and their wives abcd respectively.  Then the following crossings are necessary:

Left bank	River	Island	River	Right bank
ABCDabcd	-	-	-	-
ABCDcd	ab ®	-	-	-
ABCDcd	¬ b	a	-	-
ABCDd	bc ®	a	-	-
ABCDd	¬ c	ab	-	-
CDcd	AB ®	ab	-	-
CDcd	-	AB	ab ®	-
CDcd	-	AB	¬ b	a
CDcd	-	b	AB ®	a
CDcd	-	b	¬ B	Aa
CDcd	¬ B	b	-	Aa
BCD	cd ®	b	-	Aa
BCD	¬ d	bc	-	Aa
Dd	BC ®	bc	-	Aa
Dd	-	bc	BC ®	Aa
Dd	-	bc	¬ a	ABC
Dd	-	c	ab ®	ABC
Dd	-	c	¬ C	ABab
Dd	¬ C	c	-	ABab
d	CD ®	c	-	ABab
d	-	c	CD ®	ABab
d	-	c	¬ b	ABCDa
d	-	-	bc ®	ABCDa
d	-	-	¬ c	ABCDab
d	¬ c	-	-	ABCDab
-	cd ®	-	-	ABCDab
-	-	-	cd ®	ABCDab
-	-	-	-	ABCDabcd

16.     Getting The Nuggets Across

Let Smith, Jones and Brown be designated by the numbers 3, 2 and 1 respectively and their respective properties by (3), (2) and (1) .  In any group on either bank the total of the numbers in parentheses must never exceed the total of the plain numbers.  One solution is as follows:

Left bank	River	Right bank
123(1)(2)(3)	-	-
13(1)(3)	2(2)®	-
13(1)(3)	¬ 2	(2)
12(3)	3(1)®	(2)
12(3)	¬ 3	(1)(2)
3(3)	12 ®	(1)(2)
3(3)	¬ 2(2)	1(1)
2(2)	3(3)®	1(1)
2(2)	¬ 1(1)	3(3)
(1)(2)	12 ®	3(3)
(1)(2)	¬ 3	12(3)
(1)	3(2)®	12(3)
(1)	¬ 1	23(2)(3)
-	1(1)®	23(2)(3)
-	-	123(1)(2)(3)

A minimum of thirteen trips.

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