Solve PE 932: 2025

Author: NonlinearFruit

README.md

@@ -5,3 +5,7 @@
 </div>
 
 # [Project Euler](https://projecteuler.net)
+
+- [required] pdm <https://github.com/pdm-project/pdm>
+- `pdm install`
+- `pdm test`

project_euler/test_pe932_2025.py

@@ -0,0 +1,91 @@
+# <https://projecteuler.net/problem=932>
+# <p>For the year $2025$</p>
+# $$2025 = (20 + 25)^2$$
+# <p>Given positive integers $a$ and $b$, the concatenation $ab$ we call a $2025$-number if $ab = (a+b)^2$.<br>
+# Other examples are $3025$ and $81$.<br>
+# Note $9801$ is not a $2025$-number because the concatenation of $98$ and $1$ is $981$.</p>
+# 
+# <p>
+# Let $T(n)$ be the sum of all $2025$-numbers with $n$ digits or less. You are given $T(4) = 5131$.</p>
+# 
+# <p>
+# Find $T(16)$.</p>
+# 
+# Notes:
+# - 0 does not count of b, 100 being (10 + 0)^2 is not included
+# - Iterating over the roots from 2 to 10**8 should work
+
+# Fiddling with math 
+# - Let x be the square number (eg: 2025)
+# - Let y be the root of a square y (eg: 45)
+# - Let a be the left hand side of the root sum (eg: 20)
+# - Let b be the right hand side of the root sum (eg: 25)
+# - Given x = y^2
+# - Given y = a + b
+# - Given $"($a)($b)" == $"($y)"
+# - The number of decimal digits in a number is floor(log b) + 1
+# - Let digit_count_of(n): floor(log n) + 1
+# - Then x = a*10^digit_count_of(b) + b
+# - So b = x - a*10^digit_count_of(b)
+# - Substituting gives y = a + x - a*10^digit_count_of(b)
+# - y - x = a - a*10^digit_count_of(b)
+# - x - y = a*10^digit_count_of(b) - a
+# - x - y = a(10^digit_count_of(b) - 1)
+# - a = (x - y) / (10^digit_count_of(b) - 1)
+# - Since the digit count of b must be a whole number, 10^digit_count_of(b) - 1 must be 1 less than a power of ten
+# - x - y must be divisible by 9 or 99 or 999 etc
+# - Therefore x - y must be divisible by 9
+
+import pytest
+from math import isqrt
+
+def sum_of_2025_numbers_with_digit_count_less_than_or_equal_to(digit_count):
+    max = isqrt(10 ** digit_count)
+    return sum(
+            map(lambda tuple: tuple[2],
+                filter(lambda tuple: tuple != None,
+                       map(extract_a_and_b,
+                           [root for root in range(2, max + 1)]))))
+
+def test_sum_works_with_example():
+    assert 5131 == sum_of_2025_numbers_with_digit_count_less_than_or_equal_to(4)
+    # assert 587549 == sum_of_2025_numbers_with_digit_count_less_than_or_equal_to(6)
+    # assert 176339975 == sum_of_2025_numbers_with_digit_count_less_than_or_equal_to(8)
+
+@pytest.mark.skip(reason="lots of seconds to solve")
+def test_sum_works_with_real_number():
+    assert 72673459417881349 == sum_of_2025_numbers_with_digit_count_less_than_or_equal_to(16)
+
+def extract_a_and_b(root):
+    square = root**2
+    diff = square - root
+    if diff % 9 != 0:
+        return None
+    str_square = str(square)
+    for digits_in_b in range(1,9):
+        base = 10 ** digits_in_b
+        b = square % base
+        if b == 0:
+            continue
+        a, a_remainder = divmod(diff, base - 1)
+        if a_remainder == 0 and f'{a}{b}' == str_square:
+            return a, b, square
+
+def test_a_extracting_works():
+    assert 8 == extract_a_and_b(9)[0]
+    assert 20 == extract_a_and_b(45)[0]
+    assert 30 == extract_a_and_b(55)[0]
+
+def test_b_extracting_works():
+    assert 1 == extract_a_and_b(9)[1]
+    assert 25 == extract_a_and_b(45)[1]
+
+def test_extracting_fails_when_diff_is_not_a_multiple_of_9():
+    assert None == extract_a_and_b(8)
+
+def test_extracting_fails_when_b_is_0():
+    assert None == extract_a_and_b(10)
+    assert None == extract_a_and_b(100)
+
+def test_extracting_fails_when_b_requires_leading_zeros():
+    assert None == extract_a_and_b(99)