by Bill Graham

Logic Puzzles

This puzzle makes an excellent warm-up. Begin with a grid with rows and columns assigned to the people and grades. Insert dashes to eliminate cells corresponding to the no matching initials clue.

	Allan	Bob	Carol	Dave	Florence
A	—
B		—
C			—
D				—
F					—

Next, list the clues.
(a) Allan and the C student studied together.
(b) Carol passed.
(c) The A and B students live near Carol.
(d) Allan got a higher grade than Carol.
(e) Dave did better than Carol but worse than Florence.
Note some clues provide two pieces of information. Clue (a) tells you that Allan was not the C student; put a dash in that cell. Clue (b) tells you that Carol was not the F student. Clue (c) tells you that Carol was neither the A nor the B student.

After the clues a-c, you know that Carol's grade was a D. Put a check in the cell for Carol and D.

	Allan	Bob	Carol	Dave	Florence
A	—		c
B		—	c
C	a		—
D			√	—
F			b		—

Mark out the D row, and enter clue d. The only remaining possible grade for Allan is a B.

	Allan	Bob	Carol	Dave	Florence
A	—		c
B	√	—	c	—	—
C	a		—
D	—	—	√	—	—
F	—		b		—

Both Dave and Florence received better grades than a D, and only the A and C are left in that range. That leaves Bob with the F. Florence did better than Dave and so got the A leaving Dave with the C.

	Allan	Bob	Carol	Dave	Florence
A	—		c		√
B	√	—	c	—	—
C	a		—	√
D	—	—	√	—	—
F	—	√	b		—

2. Amateur Boxing Championships

Actually, the clues are multiple. Separate and renumber.
(a) Brown weighed in at 117 lb.
(b) Grant defeated Mitchell.
(c) O'Brien lost the 145 lb. final by a knockout.
(d) Todd was the heaviest winner of all.
(e) Slater won by a knockout.
(f) Ford hurt his hand and couldn't continue in the 118 lb. final.
Assume that someone weighing in at 117 lb will compete in the 118 lb division. Otherwise, clue a is meaningless.

		Brown	Todd	Robinson	Slater	Mitchell	O'Brien	Grant	Ford
118	W		—				—		—
	L	—	—	—	e	—	—	—	√f
126	W	a	—				—		—
	L	a	—		e		—		—
145	W	a	—				—		—
	L	a	—	—	e	—	√c	—	—
hvy	W	a	√d	—	—	—	—	—	—
	L	a	—		e	—	—		—

Grant and Mitchell fought in the only division left – the 126 lb division. Grant won. Brown must have won the 118 lb division.

		Brown	Todd	Robinson	Slater	Mitchell	O'Brien	Grant	Ford
118	W	√	—	—	—	—	—	—	—
	L	—	—	—	e	—	—	—	√f
126	W	a	—	—	—	—	—	√b	—
	L	a	—	—	e	√b	—	—	—
145	W	a	—			—	—	—	—
	L	a	—	—	e	—	√c	—	—
hvy	W	a	√d	—	—	—	—	—	—
	L	a	—		e	—	—	—	—

Only Robinson and Slater are left now. Slater must have won the 145 lb division leaving Robinson to lose the heavyweight division.

		Brown	Todd	Robinson	Slater	Mitchell	O'Brien	Grant	Ford
118	W	√	—	—	—	—	—	—	—
	L	—	—	—	e	—	—	—	√f
126	W	a	—	—	—	—	—	√b	—
	L	a	—	—	e	√b	—	—	—
145	W	a	—		√	—	—	—	—
	L	a	—	—	e	—	√c	—	—
hvy	W	a	√d	—	—	—	—	—	—
	L	a	—	√	e	—	—	—	—

3. The Rectangular Grid

First off you must recognize that column positions are important rather than actual cell positions. If a number is in one of the four column, then adjacent numbers cannot be in any adjacent column. Place the numbers in columns in sequence. The number 1 can be placed in the first or second column. The other columns are just reflections. You don't care if the number is placed in the top or bottom row.

One in first column:

1

The 2 can be put in either the 3rd or 4th column.

Two in third column; three must go in first column:

1		2
3

The 4 can go in either 3rd or 4th column. Four in third column; no place to put 5:

1		2
3		4

Four in fourth column; five must go in 2nd column; six forced into 4th column; no place for 7 and 8:

1	5	2	4
3			6

Two in fourth column:

1		2

The 3 can go in either the 1st or 2nd column. Three in first column:

1			2
3

The 4 can go in either the 3rd or 4th column. Four in third column; no place to put 5:

1		4	2
3

Four in fourth column; five must go in column 2; no place to put the 6:

1	5		2
3			4

Three in second column; four must go in column 4:

1	3		2
			4

The 5 can go in either the 1st or 2nd column. Five in first column; six must go in column 3: no place for 7

1	3	6	2
5			4

Five in second column; no place to put 6:

1	3		2
	5		4

	1	2

Three in first column:

3	1		2

The 4 may go in either 3rd or 4th column. Four in third column; five must go in column 1; no way is left to assign 6, 7 and 8:

3	1	4	2
5

Four in fourth column:

3	1		2
			4

The 5 can go in either 1st or 2nd column. Five in first column; no way is left to assign 6, 7 and 8:

3	1		2
5			4

Five in second column; no way is left to assign 6:

3	1		2
	5		4

Three in second column; four must go in fourth column; only column one is left for five; six forced into column 3; seven must go in column 1; and column 3 is left for eight:

5	1	6	2
7	3	8	4

4. Police Department Vacations

The five clues really consist of many more. For example, clue a really contains four clues. Separate them to clarify. Remember that each person takes one week in the first five months and a second week in the last five months.
(a) Plunkett took his first week before Gardner.
(b) Gardner took his first week before Phelps.
(c) Plunkett took his second week after Gardner.
(d) Gardner took his second week after Phelps.
(e) The man who vacationed in March also vacationed in September.
(f) Lopez did not take his first week in March or April.
(g) Lopez did not take his second week in August or December.
(h) Lopez did not take his first week in January.
(i) The person who took vacation in January did not take vacation in August or December.
(j) Maloney took his second week before Plunkett
(k) Maloney took his second week after Lopez.
Put in the obvious information.

	Gardner	Plunkett	Maloney	Phelps	Lopez
January					h
February
March					f
April					f
May
August					g
September
October
November
December					g

Use second-order information. For example, if you take your first week before someone, you could not be taking in May. The other person could not have taken vacation in January.

	Gardner	Plunkett	Maloney	Phelps	Lopez
January	a			b	h
February				a
March					f
April		b			f
May	b	a
August	d	c	k		g
September		d
October
November				c
December	c		j	d	g

August now has just one person possible, Phelps. December has just one person possible, Plunkett. By clue i, the January vacationer cannot be Plunkett leaving only Maloney.

	Gardner	Plunkett	Maloney	Phelps	Lopez
January	a	i	√	b	h
February			&mdash
March			—		f
April		b	—		f
May	b	a	—
August	d	c	k	√	g
September		d		—
October		—		—
November		—		c
December	c	√	j	d	g

Use the unused clue e; look for the only person who has March and September open, Gardner. This step leaves Plunkett as the February vacationer and April as the month for Phelps. Lopez becomes the May vacationer. Because Maloney took the second week after Lopez, Maloney must be the November person and Lopez gets October.

	Gardner	Plunkett	Maloney	Phelps	Lopez
January	a	i	√	b	h
February	—	√	—	—	—
March	√e	—	—	—	f
April	—	b	—	√	f
May	b	a	—	—	√
August	d	c	k	√	g
September	√e	d	—	—	—
October	—	—		—	√
November	—	—	√	c
December	c	√	j	d	g

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