Mathematics in World War Three

by Dinoj Surendran

It was the middle of the Third World War, that great carnage begun by a small group of football fans who thought their team should have been awarded a penalty in a World Cup semi-final. It was the middle of the night in some remote part of Australia. It was also very calm, with absolutely no sign that a global war was going on.

Till a flash in the sky signalled that Singaporean jet 1729 had been hit by an inconveniently accurate surface-to-air missile. Officers Ping, Pong, Ball and Van de Merwe had, for health reasons, decided not to stay on the plummeting plane and jumped. A few hours later, three of them were fished out by a passing boat. The fourth, having forgotten his parachute, had long since provided a healthy snack for a bunch of now-satiated crocodiles.

Thus it was that Ping, Pong and Ball were talking with the vessel's captain when a sailor with a face that would make a warthog look beautiful popped in.

"Pardon cap'n but we have a problem."

"Yeah?"

"We're in fog. Can't see the deck. We're totally lost."

"No problem. We'll just wait till the fog lifts."

"Can't do that, cap'n. We must get to the shore before morning. We only have an hour."

"Well, what are you waiting for, you blasted barracuda? Get to land!"

"Get to land. Ah, yes. Er. Land. That's a bit difficult. We don't know where land is."

"Hey, I know you guys love the water, and don't care about terra firma, but that's no excuse for not knowing where it is. What happened?"

"The thingy-that-tells-us-in-which-direction-we're-facing has broken down. GPS tells us we're in the middle of the river and that's all we know. The river here has two parallel banks 10km apart but we only have enough fuel left for 17km."

"Alright. So we have to get to shore. I assume we're not too fussy where in shore, so long as we get there. Hmmm. Tell the navigator to go in a straight line till the fuel gives out. We should have a good chance of reaching shore then."

That was when Ping spoke up.

"Excuse me captain, but that method has an 80% chance of success."

"So what, Officer Ping? Those are pretty good odds."

"Are you sure sir? There's a 1 in 5 chance that you could be caught by the enemy in the morning. I believe we can reduce that possibility ten times, maybe even to zero."

"Really? How?"

"Maths."

"Wake up, man! This is the real world. Maths ain't got no place here."

"Sir, if you could give us ten minutes, we'll have an answer for you."

"Alright. Get cracking."

"But one question first, sir - can the ship turn a required amount and go an exact distance?"

The captain looked at the sailor, who nodded rather bemusedly. The three airmen then quickly collected some paper and a calculator from a sailor who'd been using it to calculate the amount of money he lost on bets.

"Alright Ping, I hope you know what you're doing. Coz we sure don't." began Ball.

"No problemo. See, here's a diagram. Here're the banks of the river , and that dot is us. For an example, I'll use the captain's method. The trouble with his method is that in the worst case, the direction could be parallel - or nearabouts - to the banks and we don't reach shore in time."

Pong finally spoke, "so we need to devise a path that will definitely reach shore and is less than 17km in the worst case. Only if that's not possible will we worry about methods that have a high - but not certain - possibility of success."

"Correct. Here's one method. Let's move in a circular path. Say radius 5km. Then the worst case will be about 23km."

"How come?"

"Because this is how the worst case happens. We move very close to the shore, but since we don't know we're there, we go all the way to the other shore. About 3/4 of the circumference."

"And that's the optimum radius? Can't we do better with some other radius?"

"5 km is the minimum possible radius, of course. I feel in my bones that no larger radius will give better results."

"Okay. What about a spiral? Like this?"

"I suppose that's an idea, but you tell me how to analyse it!"

Then Ball finally looked up from his doodles.

"Eureka! Let's go for a long distance in one direction, and if after a certain distance we don't get there, we turn a certain angle and keep going till we hit shore."

"I guess straight lines might work," began Ping, "they do tend to be the shortest distance between two points, though we only know one point here... but what distance? What angle?"

"That's what I'm not sure. But I'm close. Let's do an example. We move 5 km then turn 45° and move till we hit shore."

Pong banged his hand on the table. "Listen, stop leading us on this wild-goose chase. With your method, how do we know we'll reach shore? We could end up moving parallel to the shore!"

"No, you mild-mannered moron, that is exactly what my method avoids. It makes sure that when we turn, we have gone a sufficient distance to see that we can never be parallel to the edge when we turn. Have a look at this diagram. There's only one case when we might end up moving parallel to the river. That occurs if and only if our initial direction is 45° to the shore. But we move 5km so if we do go in that direction, we'll hit the shore before we turn. So that second part of the arrow I've drawn will never really be needed."

"Not bad. But what's the worst case?"

"Ah, the worst case. That would mean we almost hit the shore when we turn, but miss, and thus move almost perpendicularly to the shore till we hit the other shore. Distance: 5+10, or approximately 17.07km."

"You've done it! 17km!"

"Almost, but not quite. What if the worst case happens and we don't have enough fuel for the last 70m? Come on, let's continue. The worst-case path has two parts. Let the distance of the first path, ie that before turning, be x. And for total generality, let us take d to be the distance between the two banks. Here of course d=10km.

The first question regards finding the value of x. It must be long enough to prevent us eventually moving parallel to the shore. Thus d/2 = x sin and x=d/2sin. So we must make sure we move this distance before turning."

"Hold it. We must always move x km before turning, right? But x is in terms of d and and we don't know "

"Correct. But we will know later on. For now, let's think about the distance of the part after the turn, let's call it y.

For that we must consider how long might we have to move after turning. The worst case occurs if we're very close to the shore after x m, but then turn and go all the way to the other shore. If you look at my diagram, you see that the angle between the line whose distance is x and the shore is also --- you can verify this quite easily. Thus the angle between the line whose distance is y and the other shore is 2. Hence y=d/sin 2.

So know we know that the distance of the total path must be d/2sin + d/sin 2.

Elementary calculus can be used to minimize this value to

or approximately 1.665d. This occurs if you take to be

Or about 51.83°. A strange place for the golden ratio to turn up! (the angle can also be expressed as arccos(1/golden ratio)).

As for your question on x, we have

which is about 0.636d. So we move about 6.36km, then turn about 51.8° --- in either direction --- and we'll reach shore before 16.7km are up."

"Brilliant! Now we can really get to the shore!"

And so they did. Well, they would have if the boat hadn't sprung a leak on the way...

EPILOGUE: All the crew of the ship perished in the croc-infested waters after the leak. But the three mathematically inclined airmen escaped after agreeing to give the reptiles a week's free maths tuition in exchange for their lives...

Author's note: It is not possible to devise a path whose worst-case length is less than 15km. This problem came from "Tomorrow's math" by Ogilvy. However I misread the problem and said that we know that the boat is in the centre of the river. If you leave out this additional bit of info, then you have an unsolved problem. But if you leave it in, then I'm not sure if it is unsolved or not. I certainly haven't solved it because I have failed to show that my solution is least. But I suspect that it is least owing to the surprising appearance of the golden ratio.