# Introduction

We are approaching New Year, so today I thought to cover a puzzle in my post. A problem that recently caught my attention is Flea Circus puzzle from Project Euler so I decided to give it a try using Julia.

The post was written under Julia 1.8.4.

# Problem statement

The Flea Circus puzzle is the following challenge:

A 30×30 grid of squares contains 900 fleas, initially one flea per square. When a bell is rung, each flea jumps to an adjacent square at random usually 4 possibilities, except for fleas on the edge of the grid or at the corners). What is the expected number of unoccupied squares after 50 rings of the bell? Give your answer rounded to six decimal places.

In this post I want to go through my thought process when I tried to solve this puzzle.

# Building intuition

I usually like to build an intuition for the problem to understand what kind of solution to expect.

The important feature of our puzzle is that all fleas move independently. The two challenges we have are that not all squares on a grid are created equal (edge squares have less neighbors than interior squares) and that the process lasts for 50 periods (so we are not sure how close it is to a stationary distribution).

To build an intuition let us assume that instead all squares are equal and that 50 can be considered a long period of time. Then we can take that a flea can be in one of the 30*30/2=450 squares with equal probability. Why 450 and not 900? To see this assume that our grid was painted in black and white like a chessboard. After 50 moves flea starting on white square must be on a white square; similarly a flea starting on a black square must be on a black square.

So what is the probability that some given square is empty after 50 moves? We have 450 fleas that potentially independently can occupy it. Therefore the probability that it is not occupied by any flea is:

julia> (1 - 1 / 450) ^ 450
0.36747030733895836


(note that it is approximately exp(-1))

By linearity of expectation we can calculate that under our simplifying assumptions we will approximately have:

julia> 900 * (1 - 1 / 450) ^ 450
330.7232766050625


empty squares.

# Building simulation

The next step towards the solution is to simulate the process. Fortunately with Julia it is easy.

We will denote the location of a flea by (i, j) tuple. Here is a function that gives us a list of locations where a flea can jump to:

const N = 30
neighbors(c::Tuple{Int, Int}) =
filter!([c .+ d for d in ((1, 0), (-1, 0), (0, 1), (0, -1))]) do new_c
all(x -> 1 <= x <= N, new_c)
end


The key element of this function is removing new potential locations that are out of bounds.

Let us create a matrix, which in cell (i, j) stores the target locations for a flea:

julia> const NEI = [neighbors((i, j)) for i in 1:30, j in 1:N]
30×30 Matrix{Vector{Tuple{Int64, Int64}}}:
[(2, 1), (1, 2)]             …  [(2, 30), (1, 29)]
[(3, 1), (1, 1), (2, 2)]        [(3, 30), (1, 30), (2, 29)]
[(4, 1), (2, 1), (3, 2)]        [(4, 30), (2, 30), (3, 29)]
[(5, 1), (3, 1), (4, 2)]        [(5, 30), (3, 30), (4, 29)]
[(6, 1), (4, 1), (5, 2)]        [(6, 30), (4, 30), (5, 29)]
[(7, 1), (5, 1), (6, 2)]     …  [(7, 30), (5, 30), (6, 29)]
[(8, 1), (6, 1), (7, 2)]        [(8, 30), (6, 30), (7, 29)]
[(9, 1), (7, 1), (8, 2)]        [(9, 30), (7, 30), (8, 29)]
⋮                            ⋱
[(24, 1), (22, 1), (23, 2)]     [(24, 30), (22, 30), (23, 29)]
[(25, 1), (23, 1), (24, 2)]     [(25, 30), (23, 30), (24, 29)]
[(26, 1), (24, 1), (25, 2)]     [(26, 30), (24, 30), (25, 29)]
[(27, 1), (25, 1), (26, 2)]  …  [(27, 30), (25, 30), (26, 29)]
[(28, 1), (26, 1), (27, 2)]     [(28, 30), (26, 30), (27, 29)]
[(29, 1), (27, 1), (28, 2)]     [(29, 30), (27, 30), (28, 29)]
[(30, 1), (28, 1), (29, 2)]     [(30, 30), (28, 30), (29, 29)]
[(29, 1), (30, 2)]              [(29, 30), (30, 29)]


Now we are ready to write a simulator:

function sim(steps)
fleas = [(i, j) for i in 1:N, j in 1:N]
for _ in 1:steps
for i in eachindex(fleas)
fleas[i] = rand(NEI[fleas[i]...])
end
end
unoccupied = trues(N, N)
for c in fleas
unoccupied[c...] = false
end
return sum(unoccupied)
end


The idea of the simulation is that we initialize it with 900 fleas, each starting on a different square. Then each of them moves randomly to one of the locations it is allowed to occupy. Finally, we count the number of unoccupied squares.

Let us run the simulation 10_000 times:

julia> using Statistics

julia> using Random

julia> Random.seed!(1234);

julia> simres = [sim(50) for _ in 1:10_000]
10000-element Vector{Int64}:
331
325
332
341
319
329
333
329
⋮
333
325
319
312
319
319
327
344


Using the collected results we can compute the mean result to get an approximation of the value we are looking for:

julia> mean(simres)
330.6895


We see that the result is close to our initial computation. Let us additionally compute the 90% confidence interval for our estimator using bootstrapping:

julia> quantile([mean(rand(simres, length(simres))) for _ in 1:10_000],
[0.05, 0.95])
2-element Vector{Float64}:
330.538595
330.84301


The width of the interval shows us that it is not realistic to get the result, up to 6 decimal places, using simulation (this was probably expected).

# Solving the problem analytically

We need to solve the problem analytically. First let us build a transition probability matrix for our problem. For this we first create two helper functions that recode (i, j) position of a flea to its numeric index:

c2i(c::Tuple{Int, Int}) = c[1] + (c[2] - 1) * N
i2c(i::Int) = reverse(divrem(i - 1, N)) .+ (1, 1)


The function uses column major ordering of locations (Julia uses this approach when defining linear indexing).

Using them we can compute the TRANS transition probability matrix:

const TRANS = zeros(N^2, N^2)
for i in 1:N^2
ni = c2i.(neighbors(i2c(i)))
TRANS[i, ni] .= 1 / length(ni)
end


Let us have a peek at the TRANS matrix:

julia> TRANS
900×900 Matrix{Float64}:
0.0       0.5       0.0       0.0       …  0.0       0.0       0.0
0.333333  0.0       0.333333  0.0          0.0       0.0       0.0
0.0       0.333333  0.0       0.333333     0.0       0.0       0.0
0.0       0.0       0.333333  0.0          0.0       0.0       0.0
0.0       0.0       0.0       0.333333     0.0       0.0       0.0
0.0       0.0       0.0       0.0       …  0.0       0.0       0.0
0.0       0.0       0.0       0.0          0.0       0.0       0.0
0.0       0.0       0.0       0.0          0.0       0.0       0.0
⋮                                       ⋱
0.0       0.0       0.0       0.0          0.0       0.0       0.0
0.0       0.0       0.0       0.0          0.0       0.0       0.0
0.0       0.0       0.0       0.0          0.0       0.0       0.0
0.0       0.0       0.0       0.0       …  0.0       0.0       0.0
0.0       0.0       0.0       0.0          0.333333  0.0       0.0
0.0       0.0       0.0       0.0          0.0       0.333333  0.0
0.0       0.0       0.0       0.0          0.333333  0.0       0.333333
0.0       0.0       0.0       0.0          0.0       0.5       0.0

julia> sum(TRANS, dims=2)
900×1 Matrix{Float64}:
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
⋮
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0


The location TRANS[i, j] encodes the probability that flea starting from location i (encoded using linear index) ends up in location j. Therefore the sum of probabilities in rows are all equal to one.

Now the linear algebra magic happens, the TRANS ^ 50 matrix encodes probability that flea starting from location i ends up in location j after 50 moves:

julia> TRANS ^ 50
900×900 Matrix{Float64}:
0.0224957  0.0          0.0313408    …  0.0          0.0
0.0        0.0325608    0.0             0.0          0.0
0.0208939  0.0          0.0295701       0.0          0.0
0.0        0.02835      0.0             0.0          0.0
0.016777   0.0          0.0246964       0.0          0.0
0.0        0.0215119    0.0          …  0.0          0.0
0.0116632  0.0          0.0182525       0.0          0.0
0.0        0.0142354    0.0             1.18131e-16  0.0
⋮                                    ⋱
0.0        1.18131e-16  0.0             0.0142354    0.0
0.0        0.0          1.18131e-16     0.0          0.0116632
0.0        0.0          0.0             0.0215119    0.0
0.0        0.0          0.0          …  0.0          0.016777
0.0        0.0          0.0             0.02835      0.0
0.0        0.0          0.0             0.0          0.0208939
0.0        0.0          0.0             0.0325608    0.0
0.0        0.0          0.0             0.0          0.0224957

julia> sum(TRANS ^ 50, dims=2)
900×1 Matrix{Float64}:
0.999999999999999
0.9999999999999988
0.9999999999999997
0.9999999999999986
0.9999999999999999
0.9999999999999998
0.9999999999999991
1.0
⋮
0.9999999999999996
0.9999999999999992
0.9999999999999987
0.999999999999999
0.9999999999999992
0.999999999999999
0.9999999999999988
0.999999999999999


You can find an explanation why this works here.

Now the rest is relatively easy. 1 .- TRANS ^ 50 is a matrix, where TRANS[i, j] tells us the probability that flea starting in position i is not in position j after 50 moves. Since fleas move independently prod(1 .- TRANS^50, dims=1) gives a probability for each cell that it is not occupied by any flea. Using linearity of expectation again, we get that the expected number of unoccupied cells is sum(prod(1 .- TRANS^50, dims=1)). Therefore the answer to the Flea circus problem is (I leave out the number to encourage you to reproduce the result yourself):

round(sum(prod(1 .- TRANS^50, dims=1)), digits=6)


# What if we wanted an exact result?

The computation above was still approximate (but the precision was enough). What if we wanted an exact result? This is easy with Julia, as we can switch to rational numbers in calculations:

const TRANS_R = zeros(Rational{BigInt}, N^2, N^2)
for i in 1:N^2
ni = c2i.(neighbors(i2c(i)))
TRANS_R[i, ni] .= 1 // length(ni)
end


Now we can compute the desired result (Warning! This time the computations take much longer.):

sum(prod(1 .- TRANS_R^50, dims=1))


I give the output in Appendix (it is quite long, as expected).

Can we make it faster? Yes, let us try using sparse arrays:

using SparseArrays
const TRANS_R_SP = spzeros(Rational{BigInt}, N^2, N^2)
for i in 1:N^2
ni = c2i.(neighbors(i2c(i)))
TRANS_R_SP[i, ni] .= 1 // length(ni)
end
TRANS_R_SP50 = TRANS_R_SP
for i in 2:50
println(lpad(i, 3), ":\t", @elapsed TRANS_R_SP50 *= TRANS_R_SP)
end


It produces the following timings of consecutive multiplication iterations:

  2:    0.0100824
3:    0.0233692
4:    0.0586901
5:    0.0591864
6:    0.0859112
7:    0.1325545
8:    0.1639544
9:    0.7365413
10:    0.2168848
11:    0.3148987
12:    0.8358322
13:    0.3933508
14:    0.970132
15:    1.0085967
16:    0.6097
17:    1.1991987
18:    1.1816013
19:    1.3753546
20:    1.3954895
21:    1.4366169
22:    1.5130404
23:    2.100761
24:    1.6176329
25:    1.7713067
26:    2.4755134
27:    2.0226999
28:    2.0449991
29:    2.7817644
30:    2.1131792
31:    2.2949804
32:    2.5469316
33:    3.2688006
34:    2.4443849
35:    2.6327831
36:    2.5831119
37:    3.2663794
38:    2.6593901
39:    2.6381515
40:    2.7143964
41:    3.5960801
42:    2.6999489
43:    4.6299449
44:    2.6266897
45:    3.086147
46:    2.939075
47:    2.7210731
48:    2.7317749
49:    3.2432152
50:    2.6385131


Now we can compute the desired fraction:

sum(prod(1 .- TRANS_R_SP50, dims=1))


(the output is the same as above and is given in Appendix)

You might wonder why I have not just done TRANS_R_SP ^ 50? The reason is that by default Julia uses exponentiation by squaring algorithm, which in our case would be slower than just doing multiplication iteratively, because TRANS_R_SP has few non-zero entries.

# Conclusions

I hope you enjoyed the puzzle and the solution. I tried to show several methods and concepts that I believe are worth practicing.

The beauty of Julia is that we could perform all the computations without installing any packages.

Happy New Year 2023!

# Appendix

If you are interested in the exact fraction representing the number we are looking for it is:

1594916209119333279277969249244834228152875163677
6251627308610309974602655007725935456430277833747
5032114412680346852027982556518161560427602079062
2980363821401845440203734684558041376621491334524
8480453760106065213319719152781581944013496218303
3324026228742213240767118408273200426409457395905
3259305972353085925030632317592370199709926747679
5163965348230742219520453869528327241503916701752
7958656576031542664199300321190807312040176225678
8602649470726805576498417540870426251929085204808
8264285469645465926569179236632298386509867712457
9252520762839368710329829821669328430611063152424
2870921332144843166780913819578239875043474333183
9667493193454163578545023664993459606760025809521
7591997522415184731455074659022022666485556962907
0617036923445844608140197059224164147430976197240
4758034658658105156940633161821995308140304104240
6587496273606310576072409516632443948112270083675
0306298428123656851818688578331387144697753827806
9029546039365862220989593884819977956369097585696
1683700929149172974949556841723371780533505215087
0115928916866575781578875425874273223139721956399
3252377107173657011338169756221698332317844745294
5906977269512285867173046172846611612371016468979
4187468918934876158274492099438949089629379569523
9878173732409250236787246817768434629788637579627
9200058619463790080778631420163026731019650612511
1268430848907342258227486393684970848372825945123
8572525008101202931590047024592764202123687929423
9293706066441191608590730025192651294842915955548
3882448273671580479296429500255634403528978881202
9949166762080377636989711119696222994712054477692
7748295851792211157201373219956360350683870143774
4425973340173528025397054612843192755948015300598
3151486837292048544369405564992295438338399383403
3221757967845779443491949352210279067457628856947
3149424761755167752804873112856153016583978439095
6389469601018599011814111357828181550751733630592
9285647852389110378459837109139105618606555810427
4750921797165961018955982988002614781533387607211
2687841158587592820092616300333764788200754508942
8031201598227309926760550319028639812264350842226
1979355583215037507461648243232521085324904489891
2114487350324310057423702149306562034348670309026
4827260556403934442058014641531314741942757254650
5297512565576083893649343084784777165749033843157
8075672705432281544341939731637434394524497208358
6715319136361454011966947389800661804881779392199
5452598724653010654023561774773763329448426273176
2127728487123950059268615767776205515384681050347
2682930354944818636528289438702651565036003376349
9776749989989588510550428339370938728234765083637
0003564914060874667639696338462422162676345259675
1275071720648590704048494501963149558867491036195
6509654323020271599939059299836559666110116685830
3776518418630115292873564688436080563705411851792
0738221819896785355779305594822421758247838909124
8533712202715171922772372529480603065427422901874
5645241828077255346035338472240795595344701467370
5811031651527497641927120305515366262540186075793
3042728606646124397477644714024170294633272528701
3057297403742107218803035867691026247629686680319
1466593450859942823191637858831903635581527808634
2207803198125134676539977407874558797070966629611
1543436910359212009384891137513143441927307288181
7967196203562000885526772313318064810296030515070
4090648464303228288525575534678486313061130545055
7182314318775217110219012244104971951851660980924
6148721247250930902297601888456954156223840570863
0259111453711483312949513479676503540153160453922
0466710911081541620263571598598706210375032281725
4516515538140906492158228836747529269807835331521
4990998581607087401796496494505382887040455387517
6977853651951906123653837144237768913819460099307
8392516186863166935226756727546934468666487803975
0319010416468821343285888740607799295157102477707
9745964992868074043001196428017338290517555388563
8647956212476351528812535288386373768334287485180
2908078417450220887351250490574296981936363536725
2938208634889574221932146427924237706174330528406
8104911374742785282279509148635382358480761952587
8505078457933636084771556055037780155028492777530
7621675212947450060657413978853605276667193549777
5169905889674410693260971703726448173460403481809
9794777402987008114061024371661885849842862169066
2358297415689920055817517263083655766216471475861
6941493259814893052921624029662220713390371085533
9795234192823914567408555521568219629186832729051
9655309618863274963844411564154676537979675559148
0801925026345263736100481796804347674286372148634
7428990271541365967437153133406489639663056745098
9416616900390884198246448783248547381907611628719
2513720632520366580132277669499251719841029611658
3170051478153118268745437227091974393090650871560
2364425755074161701397590853534354770789310500593
3080157791929316886028980045046188290439853375868
9321279140644274681059284863651518203902470113424
8602553869268967021112637286606473507636351358115
2419771869509657659322010406655800749424380961625
4272426280274669755681675032138996189176309829271
4236132977283855649065029819317676326159342263774
5153790528617401308879324857674076027536622216387
1344234253163497539593166524483536878165135603181
7500135525149289374979298113270191967514666227075
5150320041151034855092839505635545400455436369725
6540828808729513611738084661155363366833096119968
0245778936008432249520715875549087183441506071939
6311307803736449361746769631380851542271852793847
9783147914374674418164558124557576734917154320193
3342964070063730673562422018418448326927198696315
3050342073717311106244878578542188782783362756275
1900650463352523522720574096063831795802999270082
0972767311411308280059813334199233707492199546200
8919128580674645236497872479623632775397610768563
5236918453037909591801412687837073255392213201391
7747150808821255245099543172913744339209893121965
7686036994782947164781118867612529467406708127103
5391021969667966652128057620557278609689837013866
2277923266690522715485755201464505456035177778325
4535174458019746219136892710010375942414807915413
6009088571155152445432059236874655653334406298353
3681384619107757011572134805285013933234109858318
4284396295637493401515044778747731833209344669773
5813035670430616835689817133944049265052368378871
9029468414051188091580345915889990258337560461459
3740964435150783292985231713200020819738405652111
2481754250293288249113999929295922655670759686820
7743958051942073748119841372931521847865284280970
9480706599853381346731691238668320067969063999707
8714906728042593890201016378036634931819879665932
2050523613729735974466755475639044211399205336085
4305418564150974507393380824360225925890368064506
2454711835198454796127288067082733681734583323641
8178615712086917206008078595177018087110465328028
9957351625182729900951801661734292599675442994645
7513824656444828310784607892014694660400412866671
6117729768394619902917958684708673053858916215619
6404877376141641255102840688280725412165727397379
8593259429759908205402736000685310849935300991478
9195731686374408951699153633312622669782142444178
9174752290827811466905181682019607129055636827212
0503785814209185183087390101011647980004827672171
5827678690909807442375347585322090739767067543053
6638588230741944247058003343749973516135803043644
1592878742687923696538883449475593338734613428432
3539763319680590185388693546799213323439985616740
8841187691707381904611361708293883027003231626933
6997275739669225350022012300876685308273575041739
0014639977982546911200908200777259174754144135042
7835959802539985154191670247515707072736513668098
9061610254363842677636114733381719083084346642144
7181843270427778030077382671080081111876013327222
0919326368772756428216259547959852098202413818359
5164322436076087193426954740975275235948836403785
6079197397398355851287498898809058621221055096005
6332082465326887829335761030292384880129797340249
2916092087922675188111453880349182894784501122370
6214504891973731065910930838495094764146467674005
0784259679169177026934662212905530127116267752577
1051552222998494220273194474332517168226195539053
9369542956785414345312134522062695640024029257727
1729072615214633266240936202028568496275232606878
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