Cousin Sam’s 15 Challenge

Uncle Will drove in from the tree farm to drop off Alex’s cousin, Sam, so he could go to the Homeschool Math Carnival.

“Hey, Sam,” Alex said. “What’s in the sack?”

Sam smiled. “A secret puzzle.”

“Aw, c’mon,” Leon whined. “We’ll be busy with our own games at the carnival. Can’t you show us now?”

The 15 Puzzle

Sam tipped the brown paper grocery bag and spilled the contents onto the kitchen table. He had an empty picture frame and 15 rectangles cut from mat board, each with a number on one side and part of a National Geographic photo glued to the other.

“Oh, I’ve seen puzzles like this before,” Alex picked up the pieces and fit them into the frame. “We try to put the numbers in order by sliding them around.”

“Right,” Sam said. “You can slide a piece into the empty space, but you can’t turn them or lift them or jump them like checkers.”

Leon worked at the puzzle for a few minutes. “There, I got it! Now I’ll mix them up.” He flipped the pieces over. “Let’s see if Alex can put together your picture.”

“If you put the pieces in at random, there’s only a 50% chance she can solve it,” Sam said. “There are more than a trillion ways to mix the pieces, but half of those arrangements are impossible to solve.”

He patted his pocket. “I brought a list of puzzles that work. Otherwise, I’d have to start with the pieces in the right places and then mix by sliding them around, in order to be sure a player could get them back in order.”

A Puzzle for You

American puzzlemeister Sam Loyd once boasted:

In the early 1870s I made the whole world rack its brain over a tray of movable counters, that came to be known as the Fifteen puzzle. The fifteen counters were arranged in order in the tray with only 14 and 15 counters inverted. The puzzle was to get the counters into the normal arrangement by individually sliding them so that the 14 and 15 were permuted.

The $1000 reward offered for the correct solution remained unretrieved although everybody was busy on it. Funny stories were told of shop-keepers who forgot for this reason to open their shops, of respectful officials who stood throughout the night under a street lamp seeking a way to solve it. Nobody wanted to give up as everyone was confident of imminent success. It was said that navigators allowed their ships to run aground, engine drivers took their trains past stations, and farmers neglected their ploughs.

— quoted by Yakov Perelman
Fun With Maths and Physics

Loyd was so confident no one could solve his 14-15 puzzle that he offered to pay the reward out of his own pocket.

  • How could he know this was a safe wager?
  • Alexandria Jones

To Be Continued…

Read all the posts from the September/October 1999 issue of my Mathematical Adventures of Alexandria Jones newsletter.

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