Bad Math, Pepsi Points, and the Greatest Plane Non-Crash Ever

In 1995 Pepsi ran a promotion where people could collect Pepsi Points and then trade them in for Pepsi stuff. A T‑shirt was 75 points, sunglasses were 175 points, and there was even a leather jacket for 1,450 points. Wearing all three at once would get you some serious ’90s cred.

The TV commercial where the company advertised the points for stuff featured someone doing exactly that. But the people making the commercial wanted to end it on some zany bit of “classic Pepsi” craziness. So, wearing the T‑shirt, shades, and leather jacket, the ad protagonist flies his Harrier Jet to school. Apparently, this military aircraft could be yours for 7 million Pepsi Points.

Excerpted from Humble Pi: When Math Goes Wrong in the Real World by Matt Parker. Buy on Amazon.

Courtesy of Riverhead Books

The joke is simple enough: They took the idea behind Pepsi Points and extrapolated it until it was ridiculous. Solid comedy writing. But then they seemingly didn’t do the math. Seven million sure does sound like a big number, but I don’t think the team creating the ad bothered to run the numbers and check that it was definitely big enough. But someone else did. At the time, each AV‑8B Harrier II Jet brought into action cost the US Marine Corps over $20 million, and thankfully, there is a simple way to convert between USD and PP: Pepsi would let anyone buy additional points for 10 cents each. Now, I’m not familiar with the market for secondhand military aircraft, but a price of $700,000 on a $20 million aircraft sounds like a good investment. As it did to John Leonard, who tried to cash in on this.

And it was not just a lame “tried.” He went all in. The promotion required that people claim their prizes with an original order form from the Pepsi Stuff catalog, trade a minimum of 15 original Pepsi Points, and include a check to cover the cost of any additional points required. Leonard did all of that. He used an original form, he collected 15 points from Pepsi products, and submitted a check for $700,008.50. The guy actually raised the money! He was serious.

Pepsi initially refused his claim, saying “the Harrier jet in the Pepsi commercial is fanciful and is simply included to create a humorous and entertaining ad.”

But Leonard was already lawyered up and ready to fight. His attorneys fired back: “This is a formal demand that you honor your commitment and make immediate arrangements to transfer the new Harrier jet to our client.” Pepsi didn’t budge. Leonard sued, and it went to court.

The case involved a lot of discussion over whether the commercial in question was obviously a joke or if someone could conceivably take it seriously. The official notes from the judge acknowledge how ridiculous this is about to become: “Plaintiff’s insistence that the commercial appears to be a serious offer requires the Court to explain why the commercial is funny. Explaining why a joke is funny is a daunting task.” But they gave it a go!

The teenager’s comment that flying a Harrier Jet to school ‘sure beats the bus’ evinces an improbably insouciant attitude toward the relative
difficulty and danger of piloting a fighter plane in a residential area, as opposed to taking public transportation. … No school would provide landing space for a student’s fighter jet, or condone the disruption the jet’s use would cause. … In light of the Harrier Jet’s well-documented function in attacking and destroying surface and air targets, armed reconnaissance and air interdiction, and offensive and defensive antiaircraft warfare, depiction of such a jet as a way to get to school in the morning is clearly not serious.

Leonard never got his jet, and Leonard v. Pepsico Inc. is now a part of legal history. I personally find it reassuring that, if I say anything that I characterize as “zany humor,” there is legal precedent to protect me from people who take it seriously. And if anyone has a problem with that, simply collect enough Parker Points for a free photo of me not caring. (Postage and handling charges may apply.)

Pepsi took active steps to protect itself from future problems and re‑released the ad with the Harrier value at 700 million Pepsi Points. I find it amazing that they did not choose this big number in the first place. It’s not like 7 million was funnier; the company just didn’t bother to do the math when choosing an arbitrary large number.

Which brings me to the larger point: As humans, we are not good at judging the size of large numbers. Our human brains are simply not wired to be good at mathematics out of the box. Don’t get me wrong, we are born with a fantastic range of number and spatial skills; even infants can estimate the number of dots on a page and perform basic arithmetic on them. We also emerge into the world equipped for language and symbolic thought. But the skills that allow us to survive and form communities do not necessarily match formal mathematics.

We were not born with any kind of ability to intuitively understand fractions, negative numbers, or the many other strange concepts developed by mathematics, but over time, your brain can slowly learn how to deal with them. We now have school systems that force students to study math, and through enough exposure, our brains can learn to think mathematically. But if those skills cease to be used, the human brain will quickly return to factory settings.

Which makes the amount of mathematics we use in our modern society both incredible and terrifying. As a species, we have learned to explore and exploit mathematics to do things beyond what our brains can process naturally. They allow us to achieve things well beyond what our internal hardware was designed for. When we are operating beyond intuition, we can do the most interesting things, but this is also where we are at our most vulnerable. A simple math mistake can slip by unnoticed but then have terrifying consequences.

Today’s world is built on mathematics: computer programming, finance, engineering … it’s all just math in different guises. So all sorts of seemingly innocuous mathematical mistakes can have bizarre consequences. My book Humble Pi: When Math Goes Wrong in the Real World is a collection of my favorite mathematical mistakes of all time. The mistakes aren’t just amusing, they’re revealing. They briefly pull back the curtain to reveal the math that is normally being performed unnoticed behind the scenes. It’s only when something goes wrong that we suddenly have a sense of how far mathematics has let us climb—and how long the drop below might be. Take the story of the Gimli Glider.

Aircraft fuel is calculated in terms of its weight, not its volume. Temperature changes can cause things to expand and contract; the actual volume fuel takes up depends on its temperature, so it’s an unreliable measurement of quantity. Weight stays the same. So when Air Canada flight 143 was taking off from Montreal on July 23, 1983, to fly to Edmonton, it had been calculated to need at least 22,300 kilograms of fuel (plus an extra 300 kilograms for taxiing, and so on).

There was still some fuel left from the flight in to Montreal, and this was measured to check how much fuel needed to be added for the next flight. Except that both the ground maintenance personnel and the flight crew performed their calculations using pounds instead of kilograms. The amount of fuel required was in kilograms, but they filled the aircraft using pounds, and 1 pound equals only 0.45 kilograms. This resulted in the aircraft taking off with approximately half as much fuel as it required to make it to Edmonton. The Boeing 767 was now going to run out of fuel mid-flight.

In an unbelievably lucky twist of fate, the aircraft, flying with a dangerously low amount of fuel, had to make a stop in Ottawa, where the fuel levels were to be double‑checked before the plane took off again. The plane landed safely, with its crew members and passengers unaware how close they had come to running out of fuel in the air. It’s a near miss that reminds us that using the wrong units can put people’s lives in danger.

But then, in an unbelievably unlucky twist of fate, the crew doing the fuel check in Ottawa made exactly the same kilogram/pound unit error, and the aircraft was allowed to take off again without nearly enough fuel.

The fuel then ran out mid-flight.

There should be several alarm bells going off as you read this story. It’s so unbelievable as to strain credulity. Surely a plane will have fuel gauges to indicate how much fuel is left. Cars have such a gauge, and if one runs out of fuel, it merely rolls to a stop and causes a mild inconvenience: You have to walk to the nearest gas station. If a plane runs out of fuel, it also rolls to a stop—but only after dropping thousands of meters (or many more thousands of feet) out of the sky. The pilots should have been able to glance at the fuel gauge and see that they were running low.

This was not some light aircraft with a dodgy fuel gauge either. It was a brand‑new Boeing 767 recently acquired by Air Canada. A brand‑new Boeing 767 … with a dodgy fuel-gauge system. The Boeing 767 was one of the first aircraft to be furnished with all manner of avionics (aviation electronics), so much of the cockpit was electronic displays. And, like most electronics, that is all great until something goes wrong.

Because of the lack of roadside assistance when you’re thousands of feet up, in aviation, redundancy is the name of the game. Airplanes need to bring their own spares. So the electronic fuel gauge was linked to sensors in the fuel tanks by two separate channels. If the two numbers coming from each tank agreed, then the fuel gauge could confidently show the current fuel level. The signals from the sensors in the tanks (one in each of the airplane’s wings) went into a fuel‑level processor, which then controlled the gauges. Except this processor was on the blink.

One flight before its disastrous trip, the Boeing 767 was sitting in Edmonton and a certified aircraft technician named Yaremko was examining the faulty fuel gauges. He found that, if he disabled one of the fuel‑sensor channels going into the processor, the gauges started working again. He deactivated the circuit breaker for that channel, labeled it with a piece of tape marked “inoperative,” and logged the problem. The aircraft could still be compliant with the Minimum Equipment List (required for the plane to be flown safely), as long as a manual fuel check was carried out. So now the fuel double‑check consisted of the gauge system with one sensor channel and someone physically measuring the amount of fuel in the tank before takeoff.

This is where everything gets unbelievably unfortunate: The disaster makes it through several checks that could have identified and solved the problem.

The plane was flown from Edmonton to Montreal by a Captain Weir, who had misunderstood a conversation with Yaremko and thought the fuel‑gauge problem was an ongoing issue and not something that had just happened. So when he handed the aircraft to Captain Pearson in Montreal, he explained that the fuel-gauge system had a problem, but that enough fuel should be in the tank to make it to Edmonton. Captain Pearson took this to mean that the cockpit fuel gauges were completely inoperative.

While this pilot‑to‑pilot conversation was happening in Montreal, a technician named Ouellet was checking the aircraft. He did not understand the note Yaremko had logged about the fuel-gauge system, so he tested it himself, which required reactivating the circuit breaker. This caused all the gauges to go blank, and Ouellet went off to order a new processor, forgetting to re‑deactivate the circuit breaker. Captain Pearson then got into the cockpit to find all the fuel gauges blank and a label on one channel circuit breaker saying “inoperative,” which is exactly what he expected from his misunderstood conversation with Captain Weir. Because of this unfortunate series of events, a pilot was now prepared to fly an aircraft with no working fuel gauge.

Of course this would have been fine, if the fuel calculations had been performed correctly. But it was the early 1980s, and Canada was still transitioning from imperial units to metric units. In fact, the new fleet of Boeing 767s were the first aircraft for Air Canada that used metric units; all other Air Canada airplanes still measured their fuel in pounds. To add to the complication, the conversion from volume to weight used the enigmatically titled factor “specific gravity.” Had it been called “pounds per liter” or “kilograms per liter,” the problem might have been avoided. But it wasn’t. So after measuring the depth of the fuel in the tank in centimeters and successfully converting that to liters, everyone then used a specific gravity of 1.77 to do the conversion: This is the number of pounds per liter for the fuel at that temperature. The correct specific gravity of kilograms per liter would have been around 0.8. And a conversion mistake was made both before takeoff in Montreal and again during the stopover in Ottawa.

So, sure enough, in mid-flight after leaving Ottawa, the plane ran out of fuel and both engines failed within minutes of each other. This resulted in an error noise Bong! that no one in the cockpit had ever heard before. I get nervous when my laptop makes a noise I’ve never heard before; I can’t imagine what it’s like when you’re flying a plane.

The major problem with both engines failing is that indeed the plane no longer has any power to fly. A smaller but still important issue is that all the new fancy electronic displays in the cockpit needed power to work, and when the engines died, all the avionics went dead too. The pilots were left only with the analog displays: a magnetic compass, a horizon indicator, one airspeed indicator, and an altimeter. Oh yeah, and the flaps and slats, which would normally control the rate and speed of descent, also used the same power, so they were dead as well.

In the one stroke of good luck, Captain Pearson was also an experienced glider pilot. This suddenly became super useful. He was able to glide the Boeing 767 over 40 miles to a disused military base airfield in the town of Gimli. It was only a 7,200‑foot runway, but Captain Pearson was able to hit the ground within 800 feet of the start of it.

In a second stroke of good luck, the front landing gear failed, causing the front of the aircraft to scrape along the ground, providing some much‑needed braking friction, and the plane came to a halt before the end of the runway—much to the relief of the people staying in tents and campers at the far end, which was now used as a drag‑racing strip.

Here’s the thing about turning off all the engines on a 767: They fly much more silently. Some people had the fright of their life when a jet airliner suddenly appeared on the disused runway, seemingly out of nowhere.

It became known as the Gimli Glider and achieved a reasonable level of fame.

It was eventually retired in 2008 and sent to an airplane scrap yard in California. An enterprising company bought some sections of its fuselage and now sells luggage tags made from the metal skin of the Gimli Glider. I guess the idea is that the aircraft was lucky to survive a dangerous situation, so having a part of the plane should bring good luck. But then again, the vast majority of airplanes don’t crash at all, so strictly speaking, this plane was bad luck.

I bought a piece of the fuselage and attached it to my laptop, which does not seem to have crashed more or less than usual.