PSE Advent Calendar 2022 (Day 22): Santa's Sleigh Route

Question

<sub>This puzzle is part of the Puzzling Stack Exchange Advent Calendar 2022. The accepted answer to this question will be awarded a bounty worth 50 reputation.

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It's difficult for Santa to visit every household in the world on Christmas Eve. As such, his chief elf has devised a genius route that maximises the efficiency of the magic sleigh, despite its seemingly random misgivings. However, as the big day approaches, Santa realises that he cannot remember most of the route! The chief elf has now retired and taken all his important documents with him to sunny Bermuda, and Santa won't be able to contact him in time...

The route consists of many stops where Santa is allowed to take a 5-minute break from slipping down chimneys, and some major checkpoints where he has elves ready to replenish the magic dust for the reindeer. He remembers the start quite well; he needs to head due south and eventually end up at the border town of Lethem, Guyana where he can take his first break. He takes more breaks at the southern-most city in Colombia before heading north to Arauquita, in the same country. He finally gets to take his first checkpoint, in Armenia. After Armenia, he needs to complete the Australian leg of the journey. He stops in the small mining village of Karara, WA on his way to another WA town, Carnarvon. From there it's a fair way to his next stop in Collarenebri before a major checkpoint in Coober Pedy.

Santa's reindeer have helped him remember most of the route after that, but where Santa's really stuck is right at the end. He's figured out he's at his final checkpoint - the beautiful Turneffe Atoll in Belize - the only problem is he has no clue how to make it back to the North Pole afterwards.

Can you help Santa remember the final leg of the route and save Christmas?

Answer 1

To get back to the North Pole, Santa should travel via...

Boundary Bay, British Columbia (Canada)
Pelly Bay, Northwest Territories (Canada)
Lac La Biche, Alberta (Canada)
Alert, Northwest Territories (Canada)

Once in Alert - which has the northernmost airport in the world - Santa will be just 520 miles (830km) south of the true North Pole, and it's straight on to home and a rest in bed!

How do we work this out? What we need to be focussing on are:

the locations' 4-letter ICAO airport codes.

Let's start with the eight stops that are mentioned consecutively, and see what these work out as - we have...

SYLT (Lethem, Guyana)
SKLT (Leticia, the southernmost city in Colombia)
SKAT (Arauquita, Colombia)
SKAR (Armenia, Colombia - not a site in the country of Armenia, note!)
[CHECKPOINT]
YKAR (Karara, Australia)
YCAR (Carnarvon, Australia)
YCBR (Collarenebri, Australia)
YCBP (Coober Pedy, Australia)
[CHECKPOINT]

Now we can deduce how the chief elf's system for sleigh navigation works:

We move from one stop to another by way of a type of word ladder, changing one letter at a time in the ICAO airport code, with the letter to be changed each time decided by cycling through the four letter positions in order (i.e. change first letter, then next time change second, then third, then fourth, then reset at this 'checkpoint' in order to start over again by changing the first).

You can see this pattern demonstrated by the known cities:

SYLT - SKLT - SKAT - SKAR - YKAR - YCAR - YCBR - YCBP

So where do we go from Turneffe Atoll? Well, if there are to be no more checkpoints, we know - or can find out or deduce - a few things that will help. Here's my step-by-step thought process until the solution became clear:

1. The ICAO airport code for Turneffe Atoll is MZBB.

2. We need one final cycle of letter changes, since there are no more checkpoints. i.e. 4 more letter changes - one for each position.

3. On common lists of ICAO airport codes (e.g. this one) there is only one other airport in the world whose code ends in 'ZBB' and can thus be 'moved to' from MZBB: CZBB in Boundary Bay, British Columbia (Canada).

4. After this move there is one valid move to an airport with a code of the format C?BB: CYBB in Pelly Bay, Northwest Territories (Canada).

5. There are, however, 13 other airports with a code of the form CY?B dotted all over Canada, and each of these has multiple options for a further move to an airport sharing the first 3 digits of its code. It seems we need to identify a final target...

6. Since we've ended up in Canada, which is in the right 'neck of the woods' for Santa to be heading home, it makes sense that perhaps we need to end up at an airport close to the North Pole. According to some websites (e.g. this one) there is a (fictional) airport called 'Santa Claus International Airport' at coordinates of 90°00'00"N and 0°00'00"E which has been ascribed the 4-letter code 'BINP'. However, since this doesn't begin with 'CY' (and doesn't exist) it's not a lot of use to us...

7. So what's the next best thing? How about the (real) airport that's geographically closest to the North Pole? A Google search reveals this to be Alert Airport in Nunavut (Canada), and - very conveniently - it has the ICAO code CYLT. This fits perfectly with what we have found so far!

8. Can we link from CYBB to CYLT? In order to do so whilst obeying the rules we've already established, we'd need an airport with the code CYLB... and it exists! Lac La Biche in Alberta (Canada).

Thus, we have our final answer...

i.e. the route must travel along the word ladder formed by the airport codes MZBB - CZBB - CYBB - CYLB - CYLT, at which point Santa is close enough to home to wrap up his day's work. It may not be the most efficient route to take, but it's one way to do it, I guess!

^{PS One final note (courtesy of the OP in comments below):}

^{Something I had overlooked myself, but by design the code for the first stop for Santa - Lethem in Guyana - is also a single change of the first letter away from the code for Alert: SYLT, from CYLT. This adds evidence for Alert - the closest airport to the North Pole - being the start point as well as the end point! Very nice.}