Improve Project Euler problem 092 solution 1 (TheAlgorithms#5703)

* Fix typos

* Improve solution

project_euler/problem_092/sol1.py

@@ -12,11 +12,14 @@
 """
 
 
+DIGITS_SQUARED = [digit ** 2 for digit in range(10)]
+
+
 def next_number(number: int) -> int:
     """
     Returns the next number of the chain by adding the square of each digit
-    to form a neww number.
-    For example if number = 12, next_number() will return 1^2 + 2^2 = 5.
+    to form a new number.
+    For example, if number = 12, next_number() will return 1^2 + 2^2 = 5.
     Therefore, 5 is the next number of the chain.
     >>> next_number(44)
     32
@@ -27,31 +30,37 @@ def next_number(number: int) -> int:
     """
     sum_of_digits_squared = 0
     while number:
-        sum_of_digits_squared += (number % 10) ** 2
+        sum_of_digits_squared += DIGITS_SQUARED[number % 10]
         number //= 10
 
     return sum_of_digits_squared
 
 
+CHAINS = {1: True, 58: False}
+
+
 def chain(number: int) -> bool:
     """
     The function generates the chain of numbers until the next number is 1 or 89.
     For example, if starting number is 44, then the function generates the
     following chain of numbers:
     44 → 32 → 13 → 10 → 1 → 1.
     Once the next number generated is 1 or 89, the function returns whether
-    or not the the next number generated by next_number() is 1.
+    or not the next number generated by next_number() is 1.
     >>> chain(10)
     True
     >>> chain(58)
     False
     >>> chain(1)
     True
     """
-    while number != 1 and number != 89:
-        number = next_number(number)
+    if number in CHAINS:
+        return CHAINS[number]
+
+    number_chain = chain(next_number(number))
+    CHAINS[number] = number_chain
 
-    return number == 1
+    return number_chain
 
 
 def solution(number: int = 10000000) -> int: