# Robot Long and Robot High

Note: supercat's answer was the closest to what I was intending. I've started a second question to ask a better way to phrase this puzzle that will allow more people to recognize the solution without the list of caveats I had to create here.

I'm making edits as holes are pointed out to narrow down the question and avoid misdirection. If this negates an answer you gave, I am sorry. I'm trying to make this a workable puzzle. Ideally, it could be told verbally without much loss.

I intend to post a separate question asking how to improve this one since it is obviously too open right now. There's a big difference between the restrictions required for posting here and the restriction required when asking "normal" people in person.

Robot Long and Robot High are tootling along via their preferred method of locomotion when they come to a gorge. They want to get to the other side and they must jump to do so.

Here's the problem, though: At it's narrowest point, the gap is just skinny enough that Robot Long can make the jump. Robot High can jump a little higher than Robot Long but it can't jump as far. If Robot High tried to make the jump, it would plummet to its doom. They are extremely accurate and precise jumpers, able to always jump the same distance at the same speed. They have only two speeds: Stopped and Moving. On each side of the gorge, there's only enough space for one robot to jump or land at a time. (The skinny point is a narrow strip jutting out from the cliff face.)

How do they get across?

Because this puzzle is not optimized, I must make several clarifications. Please help me improve this puzzle so I don't have to list off all these restrictions:

1. There are no other materials available.
2. Not even grass grows here and the robots are not engineered for digging.
3. Climbing down and back up are not an option as they are not engineered for climbing.
4. There are no bridges or other resources available.
5. Going around the gorge is not possible at it encircles the entire planet, this being the narrowest point.
6. Jumping is their only option.
7. They can not stand atop one another as that would weaken their jumping ability such that they would both plummet to their doom.
8. There are no means by which they can combine their jumping power such as holding hands (grabbers?) or tying together.
9. The distances described for their jump limits are the furthest they can jump under the best circumstances.
10. The bottom is far enough that they would be destroyed should they fall.
11. Air friction can be neglected.
12. Assume the ground speed for both robots is fixed. (The planet is large enough and they're not jumping high enough that you can assume air speed is the same as ground speed.)

Here is an incredibly precise and detailed diagram of their position and jumping abilities that I have rendered in amazing 3D if you move your head forward and backward very quickly: (not to scale)

• Can they stand on each other?
– Bob
May 7, 2015 at 19:57
• Does rotating/angling the robot effect its trajectory (as if they jump like a ball being fired from a sling shot)? Also are the only able to jump while being stationary, or can they travel forward at to a speed and jump further? May 7, 2015 at 20:10
• Stacking would not be allowed. I'll add to the OP. The distances shown are their best distances under any condition. May 7, 2015 at 20:32
• @EngineerToast Can the robots tether/interact to each other, or is the only possible interaction between them through colliding into each other? May 7, 2015 at 20:38
• @MarkN They can communicate but have no means by which to attach to each other. May 7, 2015 at 20:39

I would guess the intended answer is that...

High takes off from the cliff edge, followed some time later by Long, such that High briefly lands on Long at a point slightly less than 15m from the edge, and can then jump off from Long to complete its journey. Depending upon the relative masses of the two robots, that approach may send Long to its doom.

A potentially safer approach would be...

have High jump from a point slightly before the cliff edge just after Long has jumped from a point somewhat further back, such that they collide short of the cliff edge. If they collide at the right angle, that would direct Long back to the ground before the edge of the cliff, while High's added momentum would allow it to sail across the gap. Once that is accomplished, Long can simply do the jump itself.

• This is pretty close to the answer I intended. This and other answers have played upon the fact that the robots having the same movement speed would mean that High would have a lower ground speed and so Long could catch up to him. There's a trick, though, that doesn't require this deeper thinking. If nobody gets closer than this answer, I'll post the answer I intended, give you the tick for being the first one this close, and ask a separate question of how to improve my original text. May 8, 2015 at 12:20
• @EngineerToast: Which of my guesses did you have in mind? I actually prefer the second one, though it would have been better with the concrete numbers you specified earlier (since they would have provided a better margin to ensure that the long robot could land safely on the near side of the cliff). May 8, 2015 at 14:58
• The first guess was closer to what I had in mind. Specifically, the "land on and jump off the other robot" portion. May 8, 2015 at 15:13
• They cannot combine their jumping power in any way, and also, they cannot stand on each other. This seems to violate both rules.
– JLee
May 8, 2015 at 20:32

Well, we're tagged with "lateral-thinking" here, so..

Robot High makes it across the gorge by

leaning over the edge such that his "upward jump" is really more of a "forward jump". He jumps laterally.

• This is not what I expected but I like it. May 8, 2015 at 12:15
• I thought of this soon after the question was posted, but some commenter, I think MarkN, asked if we could angle the robots, and the OP said no.
– JLee
May 8, 2015 at 20:33
• Oh. I only read the puzzle and the massive list of rules. I didn't read any of the comments. May 9, 2015 at 0:14

Note: this answer is no longer valid as it has been established that robots are not affectionate and, therefore, cannot hug or hold hands.

First Robot:

\begin{align} 25&=v_{x_1}t_1\\ v_{y_1}&=a_g\frac{t_1}2\\ 10&=v_{y_1}\frac{t_1}2-a_g\frac{t_1^2}8=a_g\frac{t_1^2}8\\ t_1&=\frac{4\sqrt 5}{a_g} \end{align}

Second Robot:

\begin{align} 15&=v_{x_2}t_2\\ v_{y_2}&=a_g\frac{t_2}2\\ 20&=v_{y_2}\frac{t_2}2-a_g\frac{t_2^2}8=a_g\frac{t_2^2}8\\ t_2&=\frac{4\sqrt{10}}{a_g} \end{align}

Robots Hugging tightly and jumping at same time:

$$v_{x_c}=\frac{m_1v_{x_1}+m_2v_{x_2}}{m_1+m_2}\quad v_{y_c}=\frac{m_1v_{y1}+m_2v_{y2}}{m_1+m_2}$$

Assume the robots weigh the same (I can do more if you need):

\begin{align} 0&=v_{yc}t_c-a_g\frac{t_c^2}2\\&=2(\sqrt{10}+\sqrt5)t_c-a_gt_c^2\\ t_c&=\frac{2(\sqrt {10}+\sqrt 5)}{a_g}\\ v_{x_c}t_c&=\left(\frac{25}{\left(\frac{8\sqrt 5}{a_g}\right)}+\frac{15}{\left(\frac{8\sqrt{10}}{a_g}\right)}\right)\frac{2(\sqrt{10}+\sqrt 5)}{a_g}\\ v_{xc}t_c&=21.49>20 \end{align}

• This assumes they have some arms or glue or something and weigh roughly the same. The first has now been stated to be incorrect. If they weigh different weights, the game can change. May 7, 2015 at 20:48
• I have realized that I perhaps gave too many numbers. The intent was not to misdirect. I shall restate that portion. May 7, 2015 at 20:51
• This looks like a more formal version of my answer, and the problem has been clarified 8 times since then.
– JLee
May 7, 2015 at 20:51

Since high robot (H) jumps higher, it necessarily has longer hang-time than long robot (L). Thus, if it does not jump as far, it must have a lower horizontal airspeed.

Also, it is clear that the two robots must interact in some way while H is jumping or is in the air. The question rules out any other way that H could get across.

If the "jump" mechanism is simply a quick vertical booster rocket:

• Both robots would have the same horizontal airspeed as their ground speed.
• Ergo, H's ground speed must be less than L's ground speed.
• H's booster would be stronger than L's.

So, the two robots approach the cliff, with H in front of L. They are timed to collide shortly before they reach the cliff. H jumps right before the collision. The (hopefully elastic) collision increases H's horizontal speed to at least L's ground speed, and since H's hang-time is greater, this will be more than enough to cross the chasm.

L then stops abruptly, pulls back to get a running start, and jumps the chasm. Since H's horizontal airspeed is at least as large as L's normally would be, and its hang-time is greater, it will travel farther than just the minimum distance, which means there will be time for L to stop (assuming the robots can stop abruptly).

If the jump mechanism does not add a vertical boost, but merely alters the robots velocity, with H jumping at a higher angle (> 45 degrees) than L, the same mechanism would apply. Each would have a horizontal airspeed less than their ground speed (which could be the same in this case), and H's horizontal airspeed would be less than L's ground speed. So the collision would increase H's speed to at least L's ground speed, which is more than enough.

If the jump mechanism gives a boost in both horizontal and vertical direction, then this solution probably won't work.

I also looked at a solution where they both jump, timed to collide at the point where the paths cross. That would get H across safely, but I am not sure L would make it. It would be close; after the collision, L and H would be at the same height, L would be travelling at H's typical airspeed, but would not be descending quite as quickly. It would travel farther than H would have on its own trajectory.

• I just noticed this answer is essentially the same as supercat's. May 7, 2015 at 22:09

Not sure if this breaks rules, but

Robot High starts at 'Stopped' speed. This doesn't mean that he is braked however. So, Robot Long pushes Robot High toward the cliff. At a certain point, Robot High switches to Moving speed. Since Robot High is already moving, the overall speed is increased beyond his natural ability and he can clear the jump. Then Robot Long follows across.

The robots might engage in something like the very beginning of this video

'two ball experiment: putt two balls on top of eachother, touching. Then, release them together. The top-most ball will jump much higher than expected at the cost of the jump of the bottom-most ball. What works for balls could work for robots too?

Unlikely but here is my answer

As they are really precise jumpers, I suggest that the robot that can jump only 15 m start his jump
Then the other robot jump after him to meet him mid-air and bump it to the other side of the gorge.
So the second jumper transfer some of his movement to the other one.

• Then the second one may not make it across. May 7, 2015 at 20:29
• @pacoverflow except that the sum of the two length is exactly twice the gorge width. So I make the assessment that the second robot push him exactly for the five extra meters while keeping enough momentum to reach the edge
– A.D.
May 7, 2015 at 20:32
• This is a good idea except the robots travel at the same speed. That was not clarified in the original puzzle but it is in there now. Thank you for helping to improve the question. May 7, 2015 at 20:35
• @EngineerToast are you talking ground speed or movement speed?
– A.D.
May 7, 2015 at 20:38
• @EngineerToast: The horizontal velocity of the "long" robot during its jump will be about 2.35 times that of the "high" one, as evidenced by the fact that the "long" robot jump is only half as high, and consequently takes 1/sqrt(2) times as long, as the "high" robot jump. May 7, 2015 at 20:50

This answer doesn't directly contradict with rules 8 and 9, and isn't mentioned above so here's my attempt:

Robot High gets a push (or instant collision) at the very start of the jump from Robot Long. There's no connection, coordination, or anything else required. "Jumping Power" is not combined, Robot Long is just pushing/colliding. Once Robot High makes it over, Robot Long then jumps over.

Since this seems to be a planet unlike earth:

Assume that the planet is extremely windy at times (since there is nothing around to block the winds), and the robot jumps when the wind is strong enough behind him to push him further, enough to make the jump.

Alternatively:

Long stands at the edge of the cliff, and High jumps just prior to hitting long (such that he jumps over Long). Long then jumps vertically to hit High in mid air, giving him (High) a vertical boost (which will allow high to move farther). Then you hope the math works out such that High makes it across. Long will land back to where he jumped, and can jump across on his own.