perplexus.info :: Logic : Missing Clue

Missing Clue (Posted on 2004-04-27)

Logical Larry was upset to learn that his recent submission to a logic problem magazine had multiple solutions. As it turns out, the magazine editor accidentally omitted one of his clues.

Here is the clue as it was printed:

Four children (Betty, Bobby, Billy, and Brian) enter a store selling bikes, balls, and balloons, each of which come in blue, brown, or beige.
From the clues below, determine what each child purchased:

Betty did not purchase a balloon.

Bobby and Billy purchased the same color item, which was not brown. Betty and Bobby purchased the same item.

Betty thought it best to purchase a brown item.

Brian is the only one who purchased both his item and his color.

A brown ball was not purchased. Neither was a blue bike.

The missing clue should have read, "A balloon was not purchased by _________" However, Larry cannot remember what should be in the blank to make the problem solvable.

What should be in the blank?

Submitted by Jeff

Rating: 3.2857 (7 votes)

Solution: (Hide)

anyone
The published clues yield that Betty bought a brown bike, Bobby bought a beige bike, Billy bought something beige, and Brian bought something blue. Also, we know from clue 4 that Brian did not have the same item as anyone else did.
If the missing clue were to read, "A balloon was not purchased by Billy," then Billy could still have bought a bike or a ball. Brian could have bought a balloon or, if Billy had a bike, a ball, so this clue would still leave the problem indeterminate.
If the missing clue read, "A balloon was not purchased by Brian," then Brian must have purchased a ball, but Billy could still have bought a bike or a balloon. Nothing can be further deduced, so this can't be the solution either.
Thus, the only way to make the problem solvavle is by taking away more options. Note that the problem doesn't say that the blank was filled with a name. The clue should have read, "A balloon was not purchased by anyone." That means, again, that Brian must have purchased a ball, but Billy in this case must have purchased a bicycle.
With that clue, here's what everyone bought, again:
Betty: Brown bike
Bobby: Beige bike
Billy: Beige bike
Brian: Blue ball
Now, that form of the clue makes the first one a little useless. The problem could similarly have been worded to disallow two people having exactly the same item and color .. but wouldn't have been as fun, would it?

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re: answer	K Sengupta	2023-04-16 23:52:40
re: Learning	christ85clark	2021-03-30 07:38:06
Learning	William Neon	2020-12-24 01:02:53
answer	K Sengupta	2007-11-23 04:27:51
my soln.	ravi	2004-08-23 05:01:31
Blue bike	Chad	2004-06-07 12:53:22
missing clue	sanchi	2004-05-01 05:43:43
re(2): Full Solution (major spoiler)	SilverKnight	2004-04-28 12:28:47
Solution notes:	Gamer	2004-04-27 20:36:59
re: Full Solution (major spoiler)	Lisa	2004-04-27 19:02:59
re: Full Solution (major spoiler)	Alec Lanter	2004-04-27 16:42:46
re: Full Solution (major spoiler)	Sam	2004-04-27 15:16:35
Full Solution (major spoiler)	DJ	2004-04-27 15:10:14
re: Not solvable	Sam	2004-04-27 14:50:59
Not solvable	Alec Lanter	2004-04-27 14:24:08
Two options	Saka Devi	2004-04-27 12:54:58
Solution	Brian Smith	2004-04-27 12:19:11
solution	jordan	2004-04-27 11:54:26

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