Scrooge McGrinch and the Four Elves (a Christmas Puzzle)

E. Scrooge McGrinch

In the hour before the sun peeked around the mountain, a green, green figure clumped up the hill, dysphoria washing over him yet again. Scrooge McGrinch felt misunderstood. His shrink explained he suffered from hereditary hormonal imbalance, festiphobia (a fear of holidays) and affluenza (a love of money). As the 7^th generation following the historic marriage of Ebenezer Scrooge Jr and Ethylene Glycol MacGrinch, his skin still bore the same pigment as an eight-dollar bill.

[For our vast audience of molecular biologists, see DAT1, DRD2, and SOD1-/- in ‘Condition Green’, Morley et al, JAM, 2022.]

Then he heard a racket, a cacophony of four voices, and his spirits leaped in joy. Off the trail, caught in an avalanche of snow, were wedged his sworn nemeses, a quartet of bratty elves who made his pitiful life miserable.

Four Awful Elves

Boozy, Doozy, Floozy, and Woozy had awakened at 4:30, giggling and chuckling. The lads chortled at the notion of joyriding the village’s most recent resident, Bolderdash, understudy for the nine celebrity reindeer. In the dark of the mudroom, they pulled on boots, mittens, and stocking caps. Emboldened by a generous slug of wintergreen schnapps, they headed toward the barns.

Bolderdash was not thrilled to be shaken and awakened when a bridle slipped over his nose. The brightly lit factory and rail yard swarmed with activity at that early hour, so the jackanapes led their captive reindeer into the dark before climbing aboard.

They cantered down the tracks of the Polar Express and galloped into the night. At the edge of the plateau near Kringle International Airport, a landing FedEx cargo plane zoomed overhead, startling Bolderdash. He balked. He had had enough. In a fit of pique, he bucked and skidded to a halt at the edge of a precipice.

Four little figures flew over a cliff and landed up to their chins in deep snow, fortunate they didn’t set off an avalanche. They found themselves trapped in cold white stuff, unable to move. Meantime, Bolderdash sulkily stalked back to the barns, hoping for a little more sleep before roll call.

The Problem and the Proposal

A half hour passed until Scrooge McGrinch stumbled upon the naughty elves. He said, “You… You in the red hat. What are you miscreants doing?”

The elves appeared confused. Two wore red hats and two wore green hats. “In the dark, O Verdant One, we grabbed toques without noticing the colors. None of us can see what we’re wearing.”

“You brats wear colorful hats, but you don’t know the hue, do you?” said McGrinch. “Let’s play a little game.”

Scrooge McGrinch wasn’t a mean man… Well, okay, he was mean in multiple senses of the word, but he wasn’t entirely heartless. Following a modicum of smug enjoyment, he said, “If anyone can figure out the color cap you’re wearing, I’ll rush to Ski Patrol to dig you out, else I’ll call your parents and they won’t be happy.”

The elves shivered. “We’ll give it a try, Your Viridescence.”

McGrinch grimaced. “Fine. Here are the rules.”

The Puzzle

1. Four elves in a row are immovably buried up to their chins in snow.
2. Their names and positions are Boozy₁, Doozy₂, Floozy₃, and Woozy₄.
3. Two wear red hats, two wear green, but no elf knows what color he wears.
4. Each elf can see only the elf or elves directly ahead.
5. Woozy is separated by an impenetrable snowdrift.
6. None can glimpse their own hat, nor do colors reflect off the snow.
7. Elves are not permitted to discuss what they see.

Solution after the break.

And so Scrooge McGrinch promised to rescue them if any elf correctly identified the color hat he was wearing, otherwise they faced the humiliation of begging the Big Elf’s help. Which elf might deduce his hat?

The Answer is…

Let’s walk through the solution. Obviously Woozy, the rightmost elf, can’t see anyone. A moment’s reflection shows the same is true for Floozy. That leaves Boozy and Doozy.

Boozy can see Doozy and Floozy. Doozy sees only Floozy. Neither knows enough to solve the problem.

“Hold on,” you say. “McGrinch was unfair if he didn’t pose a solvable conundrum.”

Ah, but he did. This puzzle is unusual because it’s dynamic. The elves don’t know the answer until they’re working it out. The solution derives from the solving.

Suddenly Doozy cries out, “I’m green! I’m green!”

If Doozy and Floozy were wearing the same color toques, Boozy could shout out the other color and solve the puzzle. But the two ahead of him bear different colors, so he has no idea what his own might be.

Boozy’s not knowing is the key to the puzzle.

Doozy realizes if Boozy says nothing, he sees two different colors before him. In a mental leap, Doozy reasons if Boozy sees two different colors and he sees only red, then Doozy must be green.

“Well done,” said McGrinch. “It isn’t easy being green. Be right back with a rope and Ski Patrol. And, er, happy Chanukah and merry Christmas.”

And that is how Scrooge McGrinch saved the elves before Christmas.