Empty Boxes Invariant/Algorithmic Problem

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Discussion Overview

The discussion revolves around a combinatorial problem involving the placement of boxes within boxes, specifically focusing on determining the number of empty boxes after a series of placements. Participants explore the problem's requirements, variables, and mathematical modeling, with an emphasis on understanding the relationships between the different types of boxes involved.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents the initial problem statement, outlining the process of placing medium and small boxes into large boxes and the total count of boxes at the end.
  • Another participant suggests that the total number of empty boxes includes empty large, medium, and small boxes, prompting further exploration of how to calculate these values.
  • Several participants propose different formulations for the number of empty boxes, with variations in the use of variables L (filled large boxes), M (filled medium boxes), and S (filled small boxes).
  • There is confusion regarding the correct expressions for empty boxes, with participants questioning the logic behind their formulations and the need for clarity on the definitions of L, M, and S.
  • Some participants express uncertainty about the total number of medium and small boxes and how they relate to the overall count of empty boxes.
  • Trial and error is mentioned as a method used by one participant to arrive at certain values, leading to discussions about the validity of this approach in solving the problem.
  • Participants engage in back-and-forth clarifications and corrections regarding the mathematical expressions and the relationships between the variables.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correct formulation for the total number of empty boxes or the relationships between the variables. Multiple competing views and expressions are presented, and confusion persists regarding the definitions and calculations involved.

Contextual Notes

There are unresolved assumptions about the definitions of the variables and the relationships between the different types of boxes. The discussion reflects a range of interpretations and approaches to the problem without a clear resolution.

  • #31
tiny-tim said:
that's no way to pass an exam :redface:

ii] what is the formula for the total number of all boxes (in terms of l and m), that has to be 291 ?​

(7*25) + 25 = 200
 
Last edited:
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  • #32
Which leaves enough space for 13 medium boxes to be filled with small boxes.
 
  • #33
tiny-tim said:
ii] what is the formula for the total number of all boxes (in terms of L and M), that has to be 291 ?​

SpunkyDonutz said:
(7*25) + 25 = 200

there's no L or M in that :redface:
 
  • #34
tiny-tim said:
there's no l or m in that :redface:

(7l) + 25
 
Last edited:
  • #35
SpunkyDonutz said:
(7l) + 25

so far, so good :smile:

what about M?​
 
  • #36
tiny-tim said:
so far, so good :smile:

what about m?​

7l + 7m + 25
 
  • #37
SpunkyDonutz said:
7l + 7m + 25

:biggrin:Woohoo!:biggrin:​

ok, if 7L + 7M + 25 = 291, what can you say about L and M ? :wink:
 
  • #38
tiny-tim said:
:biggrin:Woohoo!:biggrin:​

ok, if 7L + 7M + 25 = 291, what can you say about L and M ? :wink:

7L + 7M + 25 = 291
= {-25 from each side}
7L + 7M = 266
= {let L be 18}
(7*18) + 7M = 266
= {Simplify}
126 + 7M = 266
= {-126 from each side}
7M = 140
= {140 / 7 = M}
M = 20

L = 18
M = 20
 
  • #39
SpunkyDonutz said:
{let L be 18}

why? :confused:
 
  • #40
tiny-tim said:
why? :confused:

It works, I have no clue on how to solve two variable equations.
Couldn't find anything helpful on the net either.
 
  • #41
remember: you know t = 7L + 7M + 25

and you know e = 6L + 6M + 25

so what you can you say about the relation between t and e ?​
 
  • #42
tiny-tim said:
remember: you know t = 7L + 7M + 25

and you know e = 6L + 6M + 25

so what you can you say about the relation between t and e ?​

t - e is the invariant.
 
  • #43
SpunkyDonutz said:
t - e is the invariant.

hmm … i need to remind myself of the original question …
SpunkyDonutz said:
1. Introduce the variables e and t for the number of empty and the number of
total boxes, respectively.
2. Identify the information that is given about the initial and final values of e and
t.
3. Model the process of putting seven boxes inside a box as an assignment to e
and t.
4. Calculate an invariant of the assignment.
5. Combine the previous steps to deduce the final value of e

i honestly don't know what an "invariant" is in this context :redface:, so i can't say whether you're right or not

but t - e is certainly very important!

can you solve t - e ? :wink:​
 
  • #44
tiny-tim said:
i honestly don't know what an "invariant" is in this context :redface:, so i can't say whether you're right or not

An expression that doesn't change.

tiny-tim said:
but t - e is certainly very important!

can you solve t - e ? :wink:​

The calculation would be:
(6(L + M) + 25) - (7(L + M) + 25)

But I don't know what the outcome would be.
 
  • #45
remember: you know t = 7L + 7M + 25

and you know e = 6L + 6M + 25

so what is t - e ?​
 
  • #46
tiny-tim said:
remember: You know t = 7l + 7m + 25

and you know e = 6l + 6m + 25

so what is t - e ?​

m + l
 
  • #47
SpunkyDonutz said:
m + l

ok, and you know 7M + 7L + 25 = 291

sooo … ?​
 
  • #48
tiny-tim said:
ok, and you know 7m + 7l + 25 = 291

sooo … ?​

l + m = 38?
 
  • #49
yes of course :smile:

and you should be able to solve the problem now​
 
  • #50
253 empty boxes

L = 18
M = 20

Thanks :D
 
  • #51
SpunkyDonutz said:
253 empty boxes

L = 18
M = 20

Thanks :D

(yes, 253 is correct, but …)

wait a moment

L doesn't have to be 18

did you use L = 18 to get that?

if so, you're missing the point
 
  • #52
tiny-tim said:
L doesn't have to be 18

Yeah, I know it's just one of the many possibilities I know of.

tiny-tim said:
did you use L = 18 to get that?

if so, you're missing the point

I did 298 - 25 - 38 to get it.
 
  • #53
SpunkyDonutz said:
I did 298 - 25 - 38 to get it.

298? 25? :confused:
 
  • #54
tiny-tim said:
298? 25? :confused:

Argh, I'm not thinking straight, bouncing between two assignments.

I meant 291 - 38.
 
  • #55
SpunkyDonutz said:
291 - 38.

ah, that's ok then :smile:

yes, t - e = L + M, and L + M = (291 - 25)/7
 
  • #56
tiny-tim said:
ah, that's ok then :smile:

yes, t - e = L + M, and L + M = (291 - 25)/7

Thanks a lot for the help, I really appreciate it! :smile:
 

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