From f86430be4c035807055821245772c313e5710127 Mon Sep 17 00:00:00 2001
From: mindaugl <mindelek@gmail.com>
Date: Sat, 5 Apr 2025 15:55:46 +0800
Subject: [PATCH 1/9] Add initial version for euler project problem 122.
---
project_euler/problem_122/__init__.py | 0
project_euler/problem_122/sol1.py | 35 +++++++++++++++++++++++++++
2 files changed, 35 insertions(+)
create mode 100644 project_euler/problem_122/__init__.py
create mode 100644 project_euler/problem_122/sol1.py
diff --git a/project_euler/problem_122/__init__.py b/project_euler/problem_122/__init__.py
new file mode 100644
index 000000000000..e69de29bb2d1
diff --git a/project_euler/problem_122/sol1.py b/project_euler/problem_122/sol1.py
new file mode 100644
index 000000000000..f88e81edbc85
--- /dev/null
+++ b/project_euler/problem_122/sol1.py
@@ -0,0 +1,35 @@
+"""
+Project Euler Problem 122: https://projecteuler.net/problem=122
+
+Efficient Exponentiation
+"""
+
+
+def solve(nums: list[int], goal: int, depth: int) -> bool:
+ if len(nums) > depth:
+ return False
+ for el in nums:
+ if el + nums[-1] == goal:
+ return True
+ nums.append(el + nums[-1])
+ if solve(nums, goal, depth):
+ return True
+ del nums[-1]
+ return False
+
+
+def solution(n: int = 200) -> int:
+ m = [0]
+ for i in range(2, n + 1):
+ max_length = 0
+ while True:
+ nums = [1]
+ max_length += 1
+ if solve(nums, i, max_length):
+ break
+ m.append(len(nums))
+ return sum(m)
+
+
+if __name__ == "__main__":
+ print(f"{solution() = }")
From 9a380384099317437e5d0c1109beb2e46a6cfaf3 Mon Sep 17 00:00:00 2001
From: mindaugl <mindelek@gmail.com>
Date: Sat, 5 Apr 2025 16:19:11 +0800
Subject: [PATCH 2/9] Add doctests and documentation for the project euler
problem 122.
---
project_euler/problem_122/sol1.py | 30 ++++++++++++++++++++++++++++++
1 file changed, 30 insertions(+)
diff --git a/project_euler/problem_122/sol1.py b/project_euler/problem_122/sol1.py
index f88e81edbc85..06bc35fcc557 100644
--- a/project_euler/problem_122/sol1.py
+++ b/project_euler/problem_122/sol1.py
@@ -2,10 +2,29 @@
Project Euler Problem 122: https://projecteuler.net/problem=122
Efficient Exponentiation
+
+It uses the fact that for rather small n, applicable for this problem, the solution
+for each number
+can be formed by increasing the largest element.
+
+References:
+- https://en.wikipedia.org/wiki/Addition_chain
+
+>>> solution(14)
+45
"""
def solve(nums: list[int], goal: int, depth: int) -> bool:
+ """
+ Checks if nums can can have a sum equal to goal, given that length of nums does
+ not exceed depth.
+
+ >>> solve([1], 2, 2)
+ True
+ >>> solve([1], 2, 0)
+ False
+ """
if len(nums) > depth:
return False
for el in nums:
@@ -19,6 +38,17 @@ def solve(nums: list[int], goal: int, depth: int) -> bool:
def solution(n: int = 200) -> int:
+ """
+ Calculates sum of smallest number of multiplactions for each number up to
+ and including n.
+
+ >>> solution(1)
+ 0
+ >>> solution(2)
+ 1
+ >>> solution(15)
+ 50
+ """
m = [0]
for i in range(2, n + 1):
max_length = 0
From 53297f7a846c5adfa2ef1cb39eaf33af7cd71e42 Mon Sep 17 00:00:00 2001
From: Maxim Smolskiy <mithridatus@mail.ru>
Date: Mon, 14 Apr 2025 21:11:49 +0300
Subject: [PATCH 3/9] Update sol1.py
---
project_euler/problem_122/sol1.py | 24 ++++++++++++++++++++++++
1 file changed, 24 insertions(+)
diff --git a/project_euler/problem_122/sol1.py b/project_euler/problem_122/sol1.py
index 06bc35fcc557..c2d523bb5e6d 100644
--- a/project_euler/problem_122/sol1.py
+++ b/project_euler/problem_122/sol1.py
@@ -3,6 +3,30 @@
Efficient Exponentiation
+The most naive way of computing n^15 requires fourteen multiplications:
+
+ n x n x ... x n = n^15.
+
+But using a "binary" method you can compute it in six multiplications:
+ n x n = n^2
+ n^2 x n^2 = n^4
+ n^4 x n^4 = n^8
+ n^8 x n^4 = n^12
+ n^12 x n^2 = n^14
+ n^14 x n = n^15
+
+<However it is yet possible to compute it in only five multiplications:
+
+n x n = n^2
+n^2 x n = n^3
+n^3 x n^3 = n^6
+n^6 x n^6 = n^{12}
+n^12 x n^3 = n^{15}
+
+We shall define m(k) to be the minimum number of multiplications to compute n^k; for example m(15) = 5.
+
+Find sum_{k = 1}^200 m(k).
+
It uses the fact that for rather small n, applicable for this problem, the solution
for each number
can be formed by increasing the largest element.
From 3ec91f356ee43d2931566d46bc3b81183cdea394 Mon Sep 17 00:00:00 2001
From: Maxim Smolskiy <mithridatus@mail.ru>
Date: Mon, 14 Apr 2025 21:13:37 +0300
Subject: [PATCH 4/9] Update sol1.py
---
project_euler/problem_122/sol1.py | 13 +++++++------
1 file changed, 7 insertions(+), 6 deletions(-)
diff --git a/project_euler/problem_122/sol1.py b/project_euler/problem_122/sol1.py
index c2d523bb5e6d..8276c21dade7 100644
--- a/project_euler/problem_122/sol1.py
+++ b/project_euler/problem_122/sol1.py
@@ -8,6 +8,7 @@
n x n x ... x n = n^15.
But using a "binary" method you can compute it in six multiplications:
+
n x n = n^2
n^2 x n^2 = n^4
n^4 x n^4 = n^8
@@ -15,13 +16,13 @@
n^12 x n^2 = n^14
n^14 x n = n^15
-<However it is yet possible to compute it in only five multiplications:
+However it is yet possible to compute it in only five multiplications:
-n x n = n^2
-n^2 x n = n^3
-n^3 x n^3 = n^6
-n^6 x n^6 = n^{12}
-n^12 x n^3 = n^{15}
+ n x n = n^2
+ n^2 x n = n^3
+ n^3 x n^3 = n^6
+ n^6 x n^6 = n^12
+ n^12 x n^3 = n^15
We shall define m(k) to be the minimum number of multiplications to compute n^k; for example m(15) = 5.
From f03cdc31d66480d6a43a980bf1a2b427d949c6b2 Mon Sep 17 00:00:00 2001
From: Maxim Smolskiy <mithridatus@mail.ru>
Date: Mon, 14 Apr 2025 21:17:15 +0300
Subject: [PATCH 5/9] Update sol1.py
---
project_euler/problem_122/sol1.py | 8 +++-----
1 file changed, 3 insertions(+), 5 deletions(-)
diff --git a/project_euler/problem_122/sol1.py b/project_euler/problem_122/sol1.py
index 8276c21dade7..3fe1e3bae3d4 100644
--- a/project_euler/problem_122/sol1.py
+++ b/project_euler/problem_122/sol1.py
@@ -29,14 +29,10 @@
Find sum_{k = 1}^200 m(k).
It uses the fact that for rather small n, applicable for this problem, the solution
-for each number
-can be formed by increasing the largest element.
+for each number can be formed by increasing the largest element.
References:
- https://en.wikipedia.org/wiki/Addition_chain
-
->>> solution(14)
-45
"""
@@ -71,6 +67,8 @@ def solution(n: int = 200) -> int:
0
>>> solution(2)
1
+ >>> solution(14)
+ 45
>>> solution(15)
50
"""
From 32811cdbcefea24181a8e226727de021fb0a2c00 Mon Sep 17 00:00:00 2001
From: Maxim Smolskiy <mithridatus@mail.ru>
Date: Mon, 14 Apr 2025 21:17:58 +0300
Subject: [PATCH 6/9] Update sol1.py
---
project_euler/problem_122/sol1.py | 2 +-
1 file changed, 1 insertion(+), 1 deletion(-)
diff --git a/project_euler/problem_122/sol1.py b/project_euler/problem_122/sol1.py
index 3fe1e3bae3d4..aa1fe6dbeece 100644
--- a/project_euler/problem_122/sol1.py
+++ b/project_euler/problem_122/sol1.py
@@ -38,7 +38,7 @@
def solve(nums: list[int], goal: int, depth: int) -> bool:
"""
- Checks if nums can can have a sum equal to goal, given that length of nums does
+ Checks if nums can have a sum equal to goal, given that length of nums does
not exceed depth.
>>> solve([1], 2, 2)
From 30b351b9d076c1c12f242d69c500d0ed393c6b8d Mon Sep 17 00:00:00 2001
From: Maxim Smolskiy <mithridatus@mail.ru>
Date: Mon, 14 Apr 2025 21:22:38 +0300
Subject: [PATCH 7/9] Update sol1.py
---
project_euler/problem_122/sol1.py | 6 +++---
1 file changed, 3 insertions(+), 3 deletions(-)
diff --git a/project_euler/problem_122/sol1.py b/project_euler/problem_122/sol1.py
index aa1fe6dbeece..55657aa53384 100644
--- a/project_euler/problem_122/sol1.py
+++ b/project_euler/problem_122/sol1.py
@@ -72,7 +72,7 @@ def solution(n: int = 200) -> int:
>>> solution(15)
50
"""
- m = [0]
+ sum = 0
for i in range(2, n + 1):
max_length = 0
while True:
@@ -80,8 +80,8 @@ def solution(n: int = 200) -> int:
max_length += 1
if solve(nums, i, max_length):
break
- m.append(len(nums))
- return sum(m)
+ sum += max_length
+ return sum
if __name__ == "__main__":
From b64e230d12c6287f9f77f87caba740e81d6800c9 Mon Sep 17 00:00:00 2001
From: Maxim Smolskiy <mithridatus@mail.ru>
Date: Mon, 14 Apr 2025 21:22:57 +0300
Subject: [PATCH 8/9] Update sol1.py
---
project_euler/problem_122/sol1.py | 3 ++-
1 file changed, 2 insertions(+), 1 deletion(-)
diff --git a/project_euler/problem_122/sol1.py b/project_euler/problem_122/sol1.py
index 55657aa53384..a68d4cde0c83 100644
--- a/project_euler/problem_122/sol1.py
+++ b/project_euler/problem_122/sol1.py
@@ -24,7 +24,8 @@
n^6 x n^6 = n^12
n^12 x n^3 = n^15
-We shall define m(k) to be the minimum number of multiplications to compute n^k; for example m(15) = 5.
+We shall define m(k) to be the minimum number of multiplications to compute n^k;
+for example m(15) = 5.
Find sum_{k = 1}^200 m(k).
From 136f6225b044d49efe77ba245e55d493d2918225 Mon Sep 17 00:00:00 2001
From: Maxim Smolskiy <mithridatus@mail.ru>
Date: Mon, 14 Apr 2025 21:25:48 +0300
Subject: [PATCH 9/9] Update sol1.py
---
project_euler/problem_122/sol1.py | 10 +++++-----
1 file changed, 5 insertions(+), 5 deletions(-)
diff --git a/project_euler/problem_122/sol1.py b/project_euler/problem_122/sol1.py
index a68d4cde0c83..cd8b1e67708c 100644
--- a/project_euler/problem_122/sol1.py
+++ b/project_euler/problem_122/sol1.py
@@ -53,7 +53,7 @@ def solve(nums: list[int], goal: int, depth: int) -> bool:
if el + nums[-1] == goal:
return True
nums.append(el + nums[-1])
- if solve(nums, goal, depth):
+ if solve(nums=nums, goal=goal, depth=depth):
return True
del nums[-1]
return False
@@ -73,16 +73,16 @@ def solution(n: int = 200) -> int:
>>> solution(15)
50
"""
- sum = 0
+ total = 0
for i in range(2, n + 1):
max_length = 0
while True:
nums = [1]
max_length += 1
- if solve(nums, i, max_length):
+ if solve(nums=nums, goal=i, depth=max_length):
break
- sum += max_length
- return sum
+ total += max_length
+ return total
if __name__ == "__main__":