Dr. Dobb's | Treasure Arrow

Treasure Arrow

Find the treasure vault hidden in the booby-trapped castle wall.

July 01, 2005
URL:http://www.drdobbs.com/parallel/treasure-arrow/184406165

July, 2005: Treasure Arrow

Dennis is a professor of computer science at New York University. His most recent books are Dr. Ecco's Cyberpuzzles (2002) and Puzzling Adventures (2005), both published by W. W. Norton. He can be contacted at [email protected].

Solution to June Dr. Ecco

Natasha, the archaeologist was back, clear, dark eyes flashing with their usual energy. "I'm beyond pretense, Dr. Ecco," she began, speaking rapidly. "This time, I just want to make a fortune. My clients have purchased a castle. One tower in the castle is very tall and dark. It's called 'Tower Booby' because the walls are known to be booby-trapped, so everyone gives it wide berth. Until now, nobody understood why an owner would put mines in his own house. The owner who did that died when his car flew off a cliff 50 feet from a highway.

"Recently, however, the new owners gave me a document that suggests that there are more than just bombs in the walls. The documents speak of a vault placed behind a panel, surrounded by booby-trapped neighboring panels. It seems that the vault was pointed to by what we have called the 'Treasure Arrow.'

"The Treasure Arrow is a very stiff pole with a very heavy arrowhead at its right end. Five elastic bands evenly placed along the length of the pole hold the Treasure Arrow up. At rest, each band is 1 meter long.

"Because of the asymmetric weight, the arrow points downward and to the right. If we (that is you) can determine where it pointed, we can find the vault without getting blown up.

Figure 1

"The trouble is that we don't know much about the elastic bands except that:

All five were made of the same material.
They were evenly spaced along the Treasure Arrow from the left side to the arrowhead.
They were vertical.
They stretched 1 centimeter per kilogram of force.
And the rightmost band stretched 60 centimeters.

"We also know that the Treasure Arrow was 10-meters-long and the right end was 10 meters from the right wall of the tower (where the vault is supposed to be). Finally, we know that the pole without the arrowhead and the arrowhead itself weighed the same, though we don't know how much. Our question to you is, how far down the wall was the Treasure Arrow pointing?"

"Let's try a warm-up," 11-year-old Tyler suggested.

Warm-up: Suppose there are two elastic bands and the pole weighs 50 kilograms and the arrowhead also weighs 50 kilograms. But we don't know how far the right side goes. The center of mass of the pole is in the center. So the left and right bands together must support 100 kilograms. Further, taking the fulcrum to be at the arrowhead, the torque due to the pole is 5×50 kilogram-meters and the countertorque due to the left band must be 10×mass supported by stretch of the left band. Therefore, that mass must be 25 kilograms, so the left band supports 25 kilograms. Therefore, the right band must support 75 kilograms.

"Well done," said Natasha trying her best to be encouraging, though I've noticed that her attention often wandered when others spoke.

"Now can you solve the Booby Tower problem with five bands when you don't know the weights but you do know that the arrowhead and pole weigh the same, the bands stretch 1 centimeter per kilogram, and you know that the right arrowhead goes down 60 centimeters?"

After working on the problem for a little while, Tyler and Liane conferred and said, "Yes, we know where the arrow is pointing. We also know how much the arrowhead and the pole weigh."

Natasha reviewed their calculations. She smiled. "Yes!" she hissed.

"I'll cut you in." Then she flashed out.

What did Liane and Tyler say?

DDJ

July, 2005: Treasure Arrow

Figure 1: Arrowhead on the right points to the treasure. Can you find it?

July, 2005: Treasure Arrow

Dr. Ecco Solution

Solution to "The Pirates' Cantilever," DDJ, June 2005. These solutions are all due to Mike Birken.

(See Figure 1.) You might think that you want the lightest on top of the heaviest, but it turns out you want just the opposite. You put the 1-meter plank on the bottom, then the 2, then the 3, all the way up to the 10. The 10-meter plank extends out a little more than 10.06 meters. Here are the left starting positions relative to the dock edge. Negative values are above the dock. Zero is the edge.

10: 0.06106185941435823
9: -3.938938140585642
8: -5.070517087954062
7: -5.2557022731392475
6: -4.976290508433366
5: -4.426290508433366
4: -3.704068286211143
3: -2.8673335923335923
2: -1.953872053872054
1: -0.9909090909090909

(See Figure 2.) When several planks can lie on a single one, you can extend out to nearly 13 meters.

7: -6.216239067837533
6: -8.716239067837533
4: -8.100854452452918
3: -7.5714426877470355
2: -6.796442687747036
1: -5.887351778656127
10: -4.909090909090909
8: 4.977272727272727
9: -0.022727272727273373
5: -2.5227272727272734

Mike explains his solution to this second problem as follows: "My goal was to use the three longest planks to reach out as far as possible from the table edge. I started by treating plank 10 like a balance scale with planks 1-7 on the left side and planks 8 and 9 on the right. The largest value that can be moved from the left stack to the right stack while keeping the left stack heavier than the right is 5. I used plank 5 to counterbalance the weight of plank 8 on 9. The final stacks enabled me to push plank 10 a little more than halfway off the table. To make the solution more interesting, the stacking order of the left stack was intentionally selected to appear as precarious as possible, though mathematically stable."