// For every way to order the polygonal types, perform depth-first search,
// returning as soon as a solution is found.
func problem61() int {
polys := allFourDigitPolygons()
perms := tools.Permutations([]int{0, 1, 2, 3, 4, 5}, 6)
for _, perm := range perms {
// We will build up candidate solutions incrementally. The
// starting nodes are therefore the members of polys[perm[0]].
var frontier tools.Stack
for _, p := range polys[perm[0]] {
frontier.Push(Node{p})
}
for len(frontier) > 0 {
n := frontier.Pop().(Node)
z := len(n)
if z == 6 && isAllCyclic(n) {
return tools.Sum(n...)
} else if z < 6 {
// Note that perm[z] contains the next
// polygonal type in this order.
for _, y := range polys[perm[z]] {
if isCyclic(n[z-1], y) {
child := append(append(Node{}, n...), y)
frontier.Push(child)
}
}
}
}
}
return -1 // Unreachable because there is a known solution.
}
func problem68() int {
const start = 6 // Each line cannot sum to less than 6 (1+2+3)
const end = 27 // or greater than 27 (8+9+10)
// Use a depth-first search.
var frontier tools.Stack
var solutions [][]int
// Populate the initial frontier.
for N := start; N <= end; N++ {
for i := 1; i <= 10; i++ {
r := Ring{
solution: []int{i},
used: make(map[int]bool),
N: N,
}
r.used[i] = true
frontier.Push(r)
}
}
for len(frontier) > 0 {
r := frontier.Pop().(Ring)
if len(r.solution) == 15 {
// Each solution can be described uniquely starting from the
// group of three with the numerically lowest external node and
// working clockwise. So we need to filter for solutions where
// a < min(d, f, h, j)
s := r.solution
if s[0] < tools.Min(s[3], s[6], s[9], s[12]) {
solutions = append(solutions, s)
}
}
for _, child := range successors(r) {
frontier.Push(child)
}
}
// Process solutions according to the problem spec.
var strlen = 16
var answers []int
for _, sol := range solutions {
var ss []string
for _, x := range sol {
ss = append(ss, strconv.Itoa(x))
}
s := strings.Join(ss, "")
if len(s) == strlen {
val, _ := strconv.Atoi(s)
answers = append(answers, val)
}
}
return tools.Max(answers...)
}
func problem75() int {
limit := 1500000
// A mapping from values of L to the number of right-angles triangles
// with the perimiter L.
triangles := make(map[int]int)
// Use a depth-first search to exhaust the search space, starting with
// the first Pythagorean triple.
var frontier tools.Stack
frontier.Push(Triple{3, 4, 5})
for len(frontier) > 0 {
t := frontier.Pop().(Triple)
L := tools.Sum(t...)
if L > limit {
continue
}
triangles[L]++
// We're not only interested in 'primitive triples', but
// multiples too
a, b, c := t[0], t[1], t[2]
for i := 2; ; i++ {
multiple := Triple{i * a, i * b, i * c}
if tools.Sum(multiple...) >= limit {
break
}
triangles[tools.Sum(multiple...)]++
}
for _, child := range t.children() {
frontier.Push(child)
}
}
count := 0
for _, val := range triangles {
if val == 1 {
count++
}
}
return count
}
// We are going to use a concurrent depth-first search with a worker goroutine
// pool of 4. Each goroutine will search for a solution from a different
// starting prime. As soon as a solution is found, we return from the function.
// Otherwise, we wait for all starting primes to be checked, and return an
// error.
func problem60() (int, error) {
// It's not clear how many primes to search through. Experimentation
// suggests a limit of 9000 produces the correct answer: 26033. Note
// our algorithm does not guarantee the solution is the smallest
// possible, but as a matter of fact, it is. We could verify our
// answer by raising the limit to 26033, searching exhaustively, and
// observing that no smaller solutions are found.
limit := 9000
var primes []int
for i := 0; i < limit; i++ {
if tools.IsPrime(i) {
primes = append(primes, i)
}
}
c := newCache()
ans := make(chan int) // Used to send the answer.
done := make(chan bool) // Used to signal that all worker goroutines are done.
pchan := make(chan int) // Used to send worker goroutines a starting prime to search.
var wg sync.WaitGroup
// Woker goroutine pool of 4.
for i := 0; i < 4; i++ {
wg.Add(1)
go func() {
for {
p, ok := <-pchan
if !ok {
wg.Done()
return
}
// Perform depth-first search starting at the given prime.
var frontier tools.Stack
frontier.Push(Node{p})
for len(frontier) != 0 {
n := frontier.Pop().(Node)
if len(n) == 5 {
ans <- tools.Sum(n...)
}
for _, prime := range primes {
child := append(append(*new(Node), n...), prime)
if prime > tools.Max(n...) && c.allConcatToPrime(child) {
frontier.Push(child)
}
}
}
}
}()
}
go func() {
for _, p := range primes {
pchan <- p
}
close(pchan)
wg.Wait() // Wait for all workers to complete their search
done <- true // before sending completion signal.
}()
for {
select {
case x := <-ans:
return x, nil
case <-done:
return -1, fmt.Errorf("problem60: no solution found with limit %v", limit)
}
}
}