Product & Sum

Follow up to Three wise men.

Solution to “three wise men”

A certain king wishes to determine which of his three wise men is the wisest. He arranges them in a circle so that they can see and hear each other and tells them that he will put a white or black spot on each of their foreheads but that at least one spot will be white. In fact all three spots are white. He then offers his favor to the one who will first tell him the color of his spot. After a while, the wisest announces that his spot is white. How does he know?

One way of visualizing the solution is to think hypothetical scenarios where some of the men have black spots in their foreheads. Lets start supposing that two of the men have black spots: then it will be obvious to the third that he has a white spot, as at least one of the three must have one.

Now suppose that only one of the men has a black spot and take the point of view of the “wisest” (quicker at least) man with a white spot. He cannot directly infer the color of his spot, as he sees one spot of each color. But, as we have seen in the last paragraph, the man in front of him with a white spot would have known the color of his spot if he (the observer) would have had a black spot in his forehead. As he didn’t speak, then he knows that the color of his spot must be white.

Finally, lets attack the full problem taking the place of the wisest man. He knows that, if he would have had a black spot in his forehead, one of the other men would have spoken, by the previously mentioned reasons. As they haven’t done that, he can conclude that the spot in his forehead must be white.

Blue eyed islanders puzzle

A similar, but more difficult, problem is the blue eyed islanders puzzle (this version was stolen from Terry Tao):

There is an island upon which a tribe resides. The tribe consists of 1000 people, 100 of which are blue-eyed and 900 of which are brown-eyed. Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, one resident can see the eye colors of all other residents but has no way of discovering his own (there are no reflective surfaces). If a tribesperson does discover his or her own eye color, then their religion compels them to commit ritual suicide at noon the following day in the village square for all to witness. All the tribespeople are highly logical, highly devout, and they all know that each other is also highly logical and highly devout. One day, a blue-eyed foreigner visits to the island and wins the complete trust of the tribe. One evening, he addresses the entire tribe to thank them for their hospitality. However, not knowing the customs, the foreigner makes the mistake of mentioning eye color in his address, mentioning in his address “how unusual it is to see another blue-eyed person like myself in this region of the world”. What effect, if anything, does this faux pas have on the tribe?

Product & sum

This is another classical problem, first proposed (apparently) by Hans Freudenthal en 1959, and also called the “Impossible” Puzzle:

Given are X and Y, two integers, greater than 1, are not equal, and their sum is less than 100. S and P are two mathematicians; S is given the sum X+Y, and P is given the product X*Y of these numbers.
P says “I cannot find these numbers.”
S says “I was sure that you could not find them.”
P says “Then, I found these numbers.”
S says “If you could find them, then I also found them.”
What are these numbers?

Hint: a computer is very useful to solve this problem.

Solution for both puzzles in the next post.

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Three wise men

An interesting traditional puzzle (this formulation was stolen from John McCarthy):

A certain king wishes to determine which of his three wise men is the wisest. He arranges them in a circle so that they can see and hear each other and tells them that he will put a white or black spot on each of their foreheads but that at least one spot will be white. In fact all three spots are white. He then offers his favor to the one who will first tell him the color of his spot. After a while, the wisest announces that his spot is white. How does he know?

400 teracycles, 200 gigabytes, 7 collisions

(Follow-up to Sumando subconjuntos – La respuesta.)

Introduction

In a previous post (and in a comment from Guille [ɡiˈʒe] 😀 ) we have seen how the pigeonhole principle implies that a set of 70 numbers in the range [1018, 1019) must have two subsets with equal sum. But this is a non-constructive proof, as it doesn’t give us the two subsets with the same sum. To rectify this omission, in this post we will see how this “sum collision” can be obtained.

Finding collisions: simple cases

We can start by defining the problem in a more general way: given a sequence of elements xi and a function f(), find two elements of the sequence, xj and xk, such that f(xj) = f(xk). In this way the sum collision problem can be reduced to finding a duplicate in the associated sequence of sums, f(xi).

Two common ways to get duplicates in a sequence are the following:

def find_duplicate_sort(g):
    sl = sorted(g)
    prev = None
    for e in sl:
        if e == prev:
            return e
        prev = e
    return None
def find_duplicate_set(g):
    es = set()
    for e in g:
        if e in es:
            return e
        es.add(e)
    return None

The first one has O(n log n) complexity and the second one has O(1) complexity if we use a hash-based set. As the set-based approach also has a lower constant, we will use this approach in the rest of this post.

This algorithm works well if the sequences to be inspected for duplicates can be fit entirely in RAM, but in this case we have seen that tens of billions of elements must be inspected to have a high probability of finding a collision. In the next section we will analyse how this restriction can be evaded.

Finding collisions in big sets

Each of the subsets to be evaluated in this problem can be encoded using 70 bits and, to allow a simple and fast sum calculation algorithm to be used, this was rounded up to 10 bytes. Then, if we want to inspect 20 billion subsets of 35 elements to get a very high probability of not wasting the calculation time, we will need 200 GB to store the whole sequence. 200 GB of data cannot be stored in the RAM of an usual machine, but it’s quite easy to store this amount of data in a hard disk nowadays.

To allow a fast hash-set based RAM collision search while keeping the bulk of the data in disk, we can take the following approach:

  1. Generate in RAM a big number of random subsets and sort them by their sums.
  2. Find a vector of numbers (to be called “pivots”) aplitting the sorted subsets vectors by their sums in approximately equal-sized segments. (The segment n will be composed by all the subsets whose sum is between pivot n-1 and pivot n).
  3. Generate in RAM a big number of random subsets and sort them by their sums.
  4. Split the sorted subset vector in segments using the previously generated pivots and append each segment to an associated segment file (for example, append segment 3 to 0003.bin).
  5. Repeat steps 3 and 4 until getting enough subsets.
  6. Check each segment file at a time for collisions.

If we choose enough pivots, the size of each segment file will be small enough to allow easily doing step 6 with a hash-based set (each segment file won’t have the same size, as the generated subsets are random; but the law of large numbers ensures that their sizes won’t be very different).

Source code & parameters

The (ugly) C code that checked for collisions can be found in the repository associated with this blog. The chosen parameters were:

  • Number of subsets: 2·1010.
  • Number of subsets in RAM: 107.
  • Number of elements in each subset: 35 (constant).
  • Number of segments: 1000.
  • Number of slots in the hash-set: 227.

Results

The first stage (segment file generation) elapsed time was approximately 41 hours, somewhat over my original estimation of 36 hours, and the segment file range ranged from 194827630 bytes to 206242980 bytes. The second stage (collision detection inside each segment file) lasted for 12-18 hours.

The output of the second stage (discarding files where no collisions were found) was:

Processing file 218...  Collision between identical elements.
 DONE in 40.754850s.
Processing file 363...  Collision between different elements!!!
  ed940f4f5710c6351a00
  a35377a5a74a03961c00
 DONE in 38.585990s.
Processing file 394...  Collision between different elements!!!
  9ab5dfa6312e090e2a00
  6558c9c8eab667233400
 DONE in 35.570039s.
Processing file 409...  Collision between different elements!!!
  2429d70f6ff500ab2c00
  423cf8c758740ac73b00
 DONE in 34.499926s.
Processing file 434...  Collision between different elements!!!
  5a8bf5a532915f213100
  496ebc281e9958713e00
 DONE in 32.610608s.
Processing file 475...  Collision between different elements!!!
  c35d3f427a7c488c0e00
  4e37132b717a287a1900
 DONE in 21.971667s.
Processing file 655...  Collision between different elements!!!
  4653023676ea8e5f1900
  6abb914e641ba45a3900
 DONE in 21.514123s.
Processing file 792...  Collision between different elements!!!
  8a4a63d85de377130600
  853d3d45b15e0c471e00
 DONE in 21.506716s.

Each set is represented as a byte string with bit number increasing when advancing through the byte string. For example, ed940f4f5710c6351a00 represents the bit string 10110111001010011111000011110010111010100000100001100011101011000101100000000000 and, consequently, the subset with indices 0, 2, 3, 5, 6, 7, 10, 12, 15, 16, 17, 18, 19, 24, 25, 26, 27, 30, 32, 33, 34, 36, 38, 44, 49, 50, 54, 55, 56, 58, 60, 61, 65, 67, 68. Its elements are

5213588008354709077 9115813602317594993
1796745334760975384 3579709154995762145
2312952310307873817 3627590721354606439
5763055694478716846 2730952213106576953
4868653895375087301 9737387704190733086
9262565481673300485 5968266991171521006
6752113930489992229 3772194655144630358
9029836747496103755 3318990262990089104
9205546877037990475 9849598364470384044
1376783969573792128 1108556560089425769
7820574110347009988 6951628222196921724
4776591302180789869 7999940522596325715
2290598705560799669 7835010686462271800
8998470433081591390 9131522681668251869
9096632376298092495 5295758362772474604
5953431042043343946 3151838989804308537
8643312627450063997 3624820335477016277
3782849199070331143

and its sum is 203743882076389458417.

In the same way, the set a35377a5a74a03961c00 has elements

5213588008354709077 9011219417469017946
3579709154995762145 3627590721354606439
5941472576423725122 4317696052329054505
2730952213106576953 5014371167157923471
9737387704190733086 9262565481673300485
5968266991171521006 5917882775011418152
5866436908055159779 9233099989955257688
3772194655144630358 3318990262990089104
9990105869964955299 2664344232060524242
1376783969573792128 1108556560089425769
7820574110347009988 9889945707884382295
7652184907611215542 8082643935949870308
4271233363607031147 6415171202616583365
6393762352694839041 2290598705560799669
7481066850797885510 5295758362772474604
5953431042043343946 9929081270845451034
7207546018039041794 3624820335477016277
3782849199070331143

and 203743882076389458417 as its sum, the same value as the previous different subset. 😀

JS puzzle

The previous post asked what happened when the JS code

(function(f){f(f)})(function(g){g(g)})

was executed. In first place we can observe that the two anonymous functions are really the same, as they only differ in the name of a dummy parameter. Let’s call this function u() and observe what happens when we apply u() over a function f():

u(f) -> f(f)

It’s clear that u() applies its argument over itself. As in this case we are applying u over itself, the result will be

u(u) -> u(u)

giving us an infinite recursion.

Sumando subconjuntos – La respuesta

Solución “no constructiva”

(Esencialmente es la misma que Guillermo presentó en un comentario del post anterior, solo difiere en la forma de acotar 🙂 )

La solución es mucho más simple de lo que podría parecer a primera vista y se basa en el el crecimiento exponencial de la cantidad de subconjuntos respecto al tamaño del conjunto “base”.

Consideremos los posibles valores que puede tomar la suma de un subconjunto de los números: como todos los elementos son positivos, la mínima suma es 0 y corresponde al subconjunto vacío, mientras que la máxima corresponde al conjunto completo. Al tener todos los elementos 19 dígitos, si llamamos S a la suma de los elementos de un subconjunto cualquiera, tendremos:

0 ≤ S < 70·1019 < 1021.

Aprovechando la relación 103 < 210, vemos que

S < 1021 < 270.

Pero, como la cantidad de subconjuntos de un conjunto de 70 elementos es exactamente 270, acabamos de probar que hay más subconjuntos que posibles valores de la suma de sus elementos. Por lo tanto es inevitable que haya al menos dos subconjuntos con el mismo valor de la suma de sus elementos.

Un punto interesante es que la solución es la misma si agregamos el requerimiento de que los conjuntos sean disjuntos. Para ver esto, supongamos que tenemos dos subconjuntos distintos A y B con la misma suma:

sum(A) = sum(B).

Si analizamos los conjuntos A\B y B\A, observaremos que

sum(A\B) = sum(A) – sum(AB)

sum(B\A) = sum(B) – sum(AB)

sum(A\B) = sum(B\A).

Como obviamente A\B y B\A no pueden compartir elementos, entonces cada par de subconjuntos distintos con igual suma tiene asociado un par de subconjuntos disjuntos con igual suma.

Una solución constructiva?

Este problema es conocido como NP-completo, por lo que no cabe esperar una solución eficiente y general. Pero podría ser que con estos valores de parámetros fuera factible encontrar una solución particular por lo que realizaremos un análisis probabilístico.

Consideremos una distribución aleatoria uniforme en el intervalo (0, 1019): claramente tendrá una densidad de probabilidad constante e igual a 10-19. Esto implica que si elegimos un número aleatoriamente de esa distribución, considerándola ahora discreta, la probabilidad sería de 10-19, lo que concuerda con nuestra intuición.

Si analizamos ahora la distribución de probabilidad de la suma de 35 variables elegidas uniformemente de dicho intervalo, tendremos una distribución de probabilidad conocida como Irwin Hall. Pero para nuestros fines podemos aproximarla por una normal con media igual a 35 veces el promedio de los números, 2.08·1020, y desviación estándar igual a la desviación estándar de los números multiplicada por la raíz cuadrada de 35, 1.57·1019. Comparando la distribución normal con un histograma obtenido mediante simulación,

Comparación de un histograma de las sumas de combinaciones de 35 números con la aproximación normal.

se observa que, aunque la coincidencia de la media es muy buena, la desviación estándar de la normal es significativamente superior a la que se observa en el histograma. Esto se debe a que, a diferencia de lo que se supone en una distribución Irwin Hall, las variables sumadas no son independientes (podemos verlo más claramente pensando en el caso de 70 variables: la suma es única, pero la distribución de Irwin Hall nos daría una desviación estándar aún mayor que en el caso de 35 variables). Como este error nos lleva a sobrestimar la dificultad de encontrar una colisión, podemos ignorar esta discrepancia por el momento, ya que nos llevará a una aproximación conservadora.

Lo que necesitaríamos ahora es calcular la probabilidad de encontrar dos combinaciones con igual suma. Aunque queda claro que las probabilidades son mayores que en el caso de que las sumas estuvieran distribuidas uniformemente en el intervalo (0, 70·1019), no es tan simple calcular un valor exacto para esta probabilidad. Pero podemos hacer una estimación suponiendo a los valores distribuidos uniformemente en el intervalo ±1σ, lo que implica descartar menos del 35% de los valores.

Aplicando la expresión derivada en el “problema del cumpleaños” para una probabilidad de colisión p:

\displaystyle n(p) \approx \sqrt{2 \cdot 1.57 \cdot 10^{19}\ln\left(\frac{1}{1-p}\right)}.

Aceptando un 95% de probabilidad de éxito en la búsqueda de una colisión:

\displaystyle n(0.95) \approx \sqrt{2 \cdot 1.57 \cdot 10^{19}\ln 20}.

\displaystyle n(0.95) \approx 9.7\cdot 10^{9}.

Tomando en cuenta que se está desechando el 30% de los puntos, podemos concluir que examinar unos 2·1010 puntos nos debería dar una probabilidad elevada de encontrar una colisión.

Si bien es bastante complejo buscar una colisión entre más de 1010 valores, no es infactible. Pero sería mejor realizar algunas pruebas preliminares para ver si el resultado obtenido es razonable.

Prueba a escala

A modo de verificación podemos realizar una simulación a escala, empleando la suma de 30 números de 7 dígitos para disminuir la carga computacional. En este caso sigue estando garantizada la existencia de una colisión, ya que

0 ≤ S < 30·107 < 109 < 230.

Aplicando la aproximación normal, la suma de 15 números distribuidos seguirá una distribución de media 7.5·107 y varianza 15·1014/12. Por lo tanto la cantidad de sumas ensayadas para tener un 95% de probabilidad de encontrar una colisión (usando solo sumas dentro de 1σ) será:

\displaystyle n(0.95) \approx \sqrt{2 \cdot 1.1 \cdot 10^7\ln\left(\frac{1}{1-0.95}\right)} \approx 8120.

O sea que la probabilidad de requerir más de unos 14000 ( = 8200 / 0.6) intentos para encontrar una colisión debería ser pequeña. Y estos valores son mucho más simples de comprobar experimentalmente que los originales, ya que puede hacerse mediante un pequeño script:


import math, random

N = 1000

NUMBERS = [4401501, 1984319, 1064736, 6495167,
           6402639, 7578056, 9301342, 6042069,
           1144375, 5680006, 7450344, 9099174,
           2011586, 8039920, 3010493, 1658370,
           5927190, 3880633, 1318068, 9594698,
           2877906, 1792394, 9120086, 7372216,
           2141103, 7256943, 6603598, 8565018,
           1698346, 3004879]

def run_until_collision():
    combs_dict = {}
    i = 0
    while True:
        comb = random.sample(NUMBERS, 15)
        s = sum(comb)
        if s in combs_dict:
            return i, comb, combs_dict[s]
        combs_dict[s] = comb
        i += 1

def show_stats():
    total = 0.0
    sq_total = 0.0
    print 'Please wait until the count reaches 100...'
    for i in range(N):
        tries, dum, dum = run_until_collision()
        total += tries
        sq_total += tries ** 2
        if i % math.ceil(N / 100.0) == 0:
            print '%d' % (i * 100.0 / N),
    print '\n\nTries statistics'
    print '  Average: %f' % (total / N)
    print '  SD: %f' % math.sqrt(sq_total / N -
                                 (total / N) ** 2)

if __name__ == '__main__':
    show_stats()

La salida obtenida al ejecutar este script (descartando indicaciones de progreso) es:

Tries statistics
  Average: 6123.049000
  SD: 3360.260149

Conclusiones

Para poder encontrar una colisión en forma explícita en el problema original deberá ser necesario buscarla entre aproximadamente 1010 valores. Como cada subconjunto con su suma ocupará en el orden de 16 bytes y una PC suele tener menos que 100 GB de RAM, sería práctico buscar un método que permita buscar colisiones sin tener todos los valores previamente analizados en memoria (otra opción sería usar datos en disco, pero es menos elegante 😀 ).

En un próximo post analizaremos como implementar una solución a este problema y veremos si pueden encontrarse resultados.