LEGO-klodsernes matematik - Vælg afdelingungetalenter.dk/.../files/aktiviteter/legoklodsernes_matematik.pdf · nat-logo Introduktion Farve Balancerende bygninger LEGO-klodsernes

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Introduktion Farve Balancerende bygninger

LEGO-klodsernes matematik

Søren [email protected]

Institut for Matematiske FagKøbenhavns Universitet

01.10.2015

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Oversigt

1 Introduktion

2 Farve

3 Balancerende bygninger

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LEGOland

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LEGO Company profile 2004

14 15

LEGO facts and figures

- It would take 40,000,000,000 LEGO bricks stacked on top of each other to reach from the Earth to the Moon.

- A LEGO set is sold across the counter somewhere in the world every 7 seconds.

- The eight robots in the LEGO Warehouse in Billund can move 660 crates of LEGO bricks an hour.

- The King’s Castle in LEGOLAND Billund cost DKK 45,000,000 to build. A total of 160 tons of crushed granite and 150 tons of boulders were used in its decoration. The castle was built in only eight months – the fastest castle construction in Danish history.

- The big Dockosaurus, part of the water ride at LEGOLAND Deutschland, is made of 90,750 LEGO bricks and weighs 262 tons.

- Children all over the world spend 5 billion hours a year playing with LEGO bricks.

- There are 102,981,500 different ways of combining six eight-stud bricks of the same colour.

- On average each person on earth owns 52 LEGO bricks.

- With an annual output of about 306 million tyres, the LEGO Company is the world’s leading tyre manufacturer.

- If all LEGO sets sold in the past 10 years were laid end to end, they would reach from London, England, to Perth, Australia.

- Visitors to www.LEGO.com spend an average of 48 minutes exploring the site.

- The largest model at LEGOLAND Windsor is a Technosaurus, which required more than one million bricks. The smallest? A pigeon on Trafalgar Square.

- When you manufacture a LEGO brick, accuracy is measured to 2/1000 of a millimetre.

14 15

LEGO facts and figures

- It would take 40,000,000,000 LEGO bricks stacked on top of each other to reach from the Earth to the Moon.

- A LEGO set is sold across the counter somewhere in the world every 7 seconds.

- The eight robots in the LEGO Warehouse in Billund can move 660 crates of LEGO bricks an hour.

- The King’s Castle in LEGOLAND Billund cost DKK 45,000,000 to build. A total of 160 tons of crushed granite and 150 tons of boulders were used in its decoration. The castle was built in only eight months – the fastest castle construction in Danish history.

- The big Dockosaurus, part of the water ride at LEGOLAND Deutschland, is made of 90,750 LEGO bricks and weighs 262 tons.

- Children all over the world spend 5 billion hours a year playing with LEGO bricks.

- There are 102,981,500 different ways of combining six eight-stud bricks of the same colour.

- On average each person on earth owns 52 LEGO bricks.

- With an annual output of about 306 million tyres, the LEGO Company is the world’s leading tyre manufacturer.

- If all LEGO sets sold in the past 10 years were laid end to end, they would reach from London, England, to Perth, Australia.

- Visitors to www.LEGO.com spend an average of 48 minutes exploring the site.

- The largest model at LEGOLAND Windsor is a Technosaurus, which required more than one million bricks. The smallest? A pigeon on Trafalgar Square.

- When you manufacture a LEGO brick, accuracy is measured to 2/1000 of a millimetre.

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Definition

Lad bn betegne antallet af sammenhængende bygninger der kankonstrueres med n 2× 4-klodser.

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Tallet 102981500

Observation12 (465 − 25) + 25 = 102981504

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Glemte bygninger!

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Korrekte antal

b1 = 1b2 = 24b3 = 1560b4 = 119580b5 = 10116403 Sekunder

b6 = 915103765 Minutter

b7 = 85747377755 Timer

b8 = 8274075616387 Uger

b9 = 816630819554486 Maneder

b10 Ar

b11 Arhundreder

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LEGO Company profile 2006

Company profile 17

! More than 400,000,000 children andadults will play with LEGO bricks thisyear.

! LEGO products are on sale in more than130 countries.

! If you built a column of about40,000,000,000 LEGO bricks, it wouldreach the moon.

! Approx. four LEGO sets are sold each sec-ond.

! The eight robots at work in the LEGObrick warehouse in Billund can move 660boxes of elements an hour.

! The King’s Castle at LEGOLAND Billundcost DKK 45,000,000 to build. The exter-nal decoration required 160 tons ofcrushed granite and 150 tons of boulders.The King’s Castle was build in only eightmonths – the fastest-built castle in Den-mark.

! The large Dockosaurus making up part ofthe water roller-coaster in LEGOLANDGermany required 90,750 LEGO ele-ments and weighs a total of 262 kg.

! The world’s children spend 5 billion hoursa year playing with LEGO bricks.

! There are 915,103,765 different ways ofcombining six eight-stud bricks of thesame colour.

! On average every person on earth has 52LEGO bricks.

! With a production of about 306 milliontyres a year, the LEGO Group is theworld’s largest tyre manufacturer.

! If all the LEGO sets sold over the past 10years were placed end to end, they wouldreach from London, England, to Perth,Australia.

! The largest model at LEGOLAND Wind-sor is the Technosaurus, which consists ofmore than one million bricks. And thesmallest? A pigeon on Trafalgar Square.

! In the manufacture of LEGO bricks themachine tolerance is as small as 0.002 mm.

Selected LEGO statistics

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Vækst

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Spørgsmal

Hvordan vokser bn?

Sætning

Der findes konstanter c ,C sa

c · 78n ≤ bn ≤ C · 177n

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Oversigt

1 Introduktion

2 Farve


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Sætning

Ethvert plant kort med sammenhængende lande kan farves medfire farver, saledes at to lande der har en grænse til fælles er farvetmed forskellige farver

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Spørgsmal

Hvad er det mindste antal farver `2×4 saledes at enhver bygningmed 2× 4-klodser kan farves med ` farver uden at to klodser derhar en side til fælles far samme farve?

Sætning

`2×2 = 5

Sætning

6 ≤ `2×4 ≤ 8

6 ≤ `2×8 ≤ 8

4 ≤ `2×2 ≤ 8

4 ≤ `1×2 ≤ 5

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Bru Frantzen, Bøhlers Nielsen, Voigt Hansen 2013

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Atanasovska, Barnholdt, Hjuler, Strunge 2014

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Oversigt

1 Introduktion

2 Farve


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J. RECREATIONAL MATHEMATICS, Vol. 12(1),1979-80

2

Figure 6

o-1I I

(Though not quite true with Lego, weassume the blocks are twice as wide asthey are high.) Hence the tower willfall if tilted through an angle fJ

-1 2fJ >fJn = tan n(n + 1)

ON "LEGO" TOWERS / 25

nTn+1 = 1: Tn_i +1.

i=O

Obviously as n becomes large, this angle becomes very small. However wemust still define this tower as stable, as it can be tilted through an angle (J < fJnand will remain standing. Use of real (as opposed to mathematical) Lego Blockslimits this kind of tower to about eight blocks in height.The four problems are:

(1) How high must a stable tower be to overhang n units, if we insist thatit lean monotonically to one side?

(2) How high must a stable tower be to overhang n units, if we relax thisrestriction?

(3) What number, Sn of stable towers of order n are there, and does theprobability xn of a tower of order n being stable decrease to zero asn increases?

(4) What number m of maximally stable towers or order n are there?, n

1. The following recursion relation can be seen immediately by takingmoments about the point A in Figure 7.

Of these only Figure 1 is not stable. We can say that Figure 4 (inter alia) has anoverhang of one unit and Figure 5 is the most stable tower, with -its center ofmass immediately above the center of the lowest block, so it must be tiltedthrough the largest angle before falling.One must be rather careful about

the definition of "small displacement":suppose one has a tower of the form inFigure 6 with n blocks in the upperpart. The center of gravity will be ata point with coordinates

(1-.!.n' 2 •

Figure

Figure 5

24

I IFigure 2

I I

Figure 4

Figure 1

ON "LEGO" TOWERS

We present four problems relating to the building toy "Lego Blocks." Two aresolved analytically, and we present numerical solutions to the other two.The building toy, "Lego Blocks," must be known to everyone: if it is

unfamiliar to the reader, we can only suggest he obtain a small set. We definea tower as being constructed from the smaller common size of blocks, 2 unitsby 2 units, with only one block on each level, connected in the obvious sense.A stable tower is one which is stable against small displacements and each blockis connected to its neighbors by at least two lugs. There are 3n -1 possible ordern towers. The five pictured below, in side view, together with mirror images ofthe first four, comprise the order 3 towers.

P. J. S. WATSONZaria, Nigeria

© 1979, Baywood Publishing Co., Inc.

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Spørgsmal

Hvordan aftager andelen af tarne bygget med 2× 1 LEGO-klodserder er maksimalt stabile i den forstand at deres massemidtpunktligger lige over centrum i den nederste klods?

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Watson

We find the following results:

1 2 3 4 5 6 7 8 9 10 11 12

1 1 1 3 7 15 35 87 217 547 1417 3735

The proportion of [...] maximally stable towers seems to go as

c

n log3(n)

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Sætning

Andelen af maksimalt stabile tarne aftager som

3

2√πn−3/2

nar n→∞.

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Spørgsmal

Hvordan aftager andelen af pyramider bygget med 2× 1LEGO-klodser der er maksimalt stabile i den forstand at deresmassemidtpunkt ligger lige over centrum i den nederste klods?

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Kode pyramider

Observation

Der er enkonkret enentydig afbildning mellem alle tarne med nklodser, og

{−, 0,+}n−1

Lad Pn være mængden af alle pyramider med n klodser.

Lemma

Der er en konkret enentydig afbildning

τ : 1{0, 1}2n−1#0=n → Pn

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Tilfælde P

1 1 0 1 0 0

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Tilfælde P

1 1 0 1 0 0

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Tilfælde P

1 1 0 1 0 0

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Tilfælde P

1 1 0 1 0 0

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Tilfælde P

1 1 0 1 0 0

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Tilfælde P

1 1 0 1 0 0

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Tilfælde PN

1 0 0 0 1 1

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Tilfælde PN

1 0 0 0 1 1

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Tilfælde PN

1 0 0 0 1 1

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Tilfælde PN

1 0 0 0 1 1

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Tilfælde P

1 0 0 0 1 1

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Tilfælde PN

1 0 0 0 1 1

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Tilfælde PN

1 0 0 0 1 1

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JOURNAL OF COMBINATORIAL THEORY 7) 130 -135 (1969)

B a l a n c i n g Equa l W e i g h t s o n t h e I n t e g e r L ine

R. E. ODEH AND E. J. COCKAYNE

Department of Mathematics, University of Victoria, Victoria, B. C., Canada

Communicated by L. Moser

Received July 4, 1968

ABSTRACT

The problem is to determine the function f(n, r) which is the total number of balance positions when r equal weights are placed on a centrally pivoted uniform rod at r distinct points whose coordinates with respect to the center as origin are a subset of the 2n -t- 1 integers {0, ~ 1, ~-2 ..... ~n}. The function is evaluated precisely for small n and estimated accurately for all n using asymptotic statistical theory.

1. INTRODUCTION

The p rob lem considered in this pape r was recent ly posed for us by L. Mose r as follows. Determine the funct ion f in, r) tha t is the number o f different posi t ions in which r equal weights can be p laced at r dist inct poin ts on a central ly p ivoted un i form rod, while equi l ibr ium is mainta ined. Fu r the rmore the coordina tes o f these points with respect to the center as origin are a subset o f the 2n q- 1 integers {0, • i 2 , . . . , i n } . Thus, for example , f (2 , 2) = 2 and f(3, 4) = 5. Due to the r ap id growth off (n , r) as n increases, the const ruct ion o f a table o f f (n , r) , say for n = 1 ..... 100, is extremely cumbersome.

In Section 2 we show how f in, r) m a y be evaluated numerical ly for a smal ler range of n using a recurrence formula. In Sect ion 3 two approximat ions for f(n, r) are derived using asympto t ic statistical theory.

Two tables are given. Table 1 gives exact values o f f in, r) for values of n = 1(I) 12, r = 1(1)n. Table 2 gives a compar i son o f the exact values o f f (n , r) with the approximat ions .

2. NUMERICAL EVALUATION OF f (n , r)

The fol lowing no ta t ion and re la t ionships have facil i tated the const ruct ion of a par t ia l table of values for f(n, r). Firs t , by an (x, y) set

130

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Theorem

x-koordinaten for massemidtpunktet af pyramiden givet ved

τ(1x−n+1 · · · x0 · · · xn−1)

er blot1

n

n−1∑k=−n+1

kxk

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Sætning

Andelen af maksimalt stabile pyramider aftager som

√3√πn−3/2

nar n→∞

Documents

LEGO-klodsernes matematik - Vælg afdelingungetalenter.dk/.../files/aktiviteter/legoklodsernes_matematik.pdf · nat-logo Introduktion Farve Balancerende bygninger LEGO-klodsernes