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2 Mathematical problems --> Weights problem and dominos problem


the maze runner
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1)You have a weight balance. You have objects of weight 1,2,3,4, to 100 units. You need to be able to measure them with minimum number of standard weights which you will weight against. For example the trivial solution is weights 1,2,3, to 100 namely 100 weights. But this can be improved upon.

2)You have a chess board 8X8 board. The top left and bottom right corners are snipped out. You have 62 remaining squares. You have 31 dominos each of which has the dimension of 2X1. Place the dominos on the board so that all squares are covered. Judge if it is possible to do this. If it is possible what is the arrangement if not possible justify why not possible.

 

I will post solutions in a few days

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Ooh! I think I know the second one because I've read about it. It's not possible. It was part of some experiment which I can't remember the details of. Sorry. I'm supposed to be using logic to solve these and it seems like cheating to say I've heard about this. But my recollection of it is so poor that I'm not completely sure I'm right.

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10 hours ago, Fluent In Silence said:

Ooh! I think I know the second one because I've read about it. It's not possible. It was part of some experiment which I can't remember the details of. Sorry. I'm supposed to be using logic to solve these and it seems like cheating to say I've heard about this. But my recollection of it is so poor that I'm not completely sure I'm right.

Yes. It is not possible. top left corner and bottom right corner squares are same color. there are then 30 light squares and 32 dark squares on the modified board. each domino of 2X1 naturally covers one dark square and one light square. hence 31 dominos in any arrangement will cover 31 light squares and 31 dark squares necessarily but since there are not 31 light squares only 30 hence it is impossible. kind of like a reasoning type of problem.

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Ok time for the solution to weights problem.

I solved this a long long time ago I don't remember how I derived the solution but the solution is that the weights are powers of 3

Like 1,3,9,27,81 etc.

Illustrating for example:-

 

1 --> 1

2--> 3-1

3-->3

4-->3+1

5-->9-(3+1)

6-->9-3

7-->(9+1)-3

8-->9-1

9-->9

10-->9+1

11-->9+3-1

12-->9+3

13-->9+3+1

14-->27-(9+3+1)

 

etc.

Hence using powers of 3 you can measure out all weights (integers) with the minimum possible weights. Btw, I have not checked the answer with the person who created the problem but I am confident this is the correct answer.

Edited by the maze runner
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  • 2 weeks later...
On 7/24/2022 at 2:12 AM, Fluent In Silence said:

No mathematicians here? I like the idea Maze Runner but I'm studying psychology and I'm not much use at these. That chess board one was used in a psychology experiment, though I can't recall what the point of it was.

psychology is also interesting especially cognitive psychology

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One more problem -->

Two trains approach each other one at 60 mph other at 40mph. They are traveling towards each other. When they are one mile apart a bird starts from 60mph train and at 100mph flies to 40 mph train. When reaches turns back and goes to 60mph train and then turns back again when reaches 60mph train and keeps doing this till the point of collision. What is the total distance covered by the bird till the point of collision?

There is a story about this problem which I will tell after I tell the solution.

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I am quite certain this is not the answer you are looking for, but: Approximately 5 feet. This is because there are no birds capable of sustained level flight at speeds approaching 100mph.* A few birds have been clocked at airspeeds exceeding 100mph, but exclusively in vertical dives; therefore, the bird in question, having exited the 60mph train at a speed of 100mph, would have therefore been in a dive, and would have impacted the track within about five feet. Further progress at that point would have been physically impossible from a splattered state.

(Another possible answer could be: Some sort of divide-by-zero error, because there can never be a collision. No railroad runs two locomotives in opposite directions on the same track, for the obvious reasons.**)

As I say, almost certainly not the correct answers, but...word problems were always my nemesis. Math hates me and the feeling is mutual, so I have personally always felt as though mixing words and math is a crime against nature anyway. 😃

 

*The White-Throated Needletail has been reported to have a top level-flight speed of 105mph, but this figure has never been verified nor published, and cannot be confirmed.
**In the early days of American railroads, public spectacles were sometimes arranged in which two locomotives were intentionally collided head-on, but this seems to have been a rare occurrence.

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Haha you get +50 points for creativity but -50 for calculation.

Now on to the answer.

Basically there are two ways to compute the answer. First way is to calculate the expression for each hop and write the series. The sum will be in the form of an infinite series which converges.

The second more clever way is to just compute the time for collision and then just multiply that number with the speed of the bird.

This question was asked of John von Neumann and he spat out the answer immediately. The questioner was like "so you got the trick immediately ha". von Neumann said "what trick I just summed up the infinite series"

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