Project Euler

A Bold Proposition

Let A be an affine plane over a radically integral local field F with residual characteristic p. We consider an open oriented line section U of A with normalized Haar measure m. Define f(m, p) as t...

Source sync Apr 19, 2026

Problem #0836

Level Level 06

Solved By 5,435

Languages C++, Python

Answer \texttt{aprilfoolsjoke

Length 341 words

analytic_mathconstructivealgebra

Let $A$ be an affine plane over a radically integral local field $F$ with residual characteristic $p$.

We consider an open oriented line section $U$ of $A$ with normalized Haar measure $m$.

Define $f(m, p)$ as the maximal possible discriminant of the jacobian associated to the orthogonal kernel embedding of $U$ into $A$.

Find $f(20230401, 57)$. Give as your answer the concatenation of the first letters of each bolded word.

Problem 836: A Bold Proposition

Mathematical Foundation

Theorem (Non-existence of Coherent Interpretation). The compound expression “radically integral local field” does not correspond to any standard mathematical object. The problem statement is intentionally constructed from plausible-sounding but mutually incompatible mathematical terminology.

Proof. A “local field” is a well-defined concept (a non-discrete, locally compact, totally disconnected topological field, e.g., $\mathbb{Q}_p$ or $\mathbb{F}_q((t))$ ). However, the modifier “radically integral” is not a recognized qualifier for local fields in any standard reference (Serre, Neukirch, Cassels-Froehlich). Similarly, while “Jacobian” and “orthogonal kernel embedding” are individually meaningful in appropriate contexts (algebraic geometry and functional analysis, respectively), their composition in this sentence does not form a coherent mathematical construction. The parameter $m = 20230401$ encodes the date April 1, 2023, confirming the problem is an April Fools’ Day puzzle. $\square$

Lemma (Extraction Rule). The answer is obtained by concatenating the first letter of each bolded word in the problem statement.

Proof. The problem explicitly instructs: “Give as your answer the concatenation of the first letters of each bolded word.” The bolded words, in order, are:

Phrase	Bolded words	First letters
affine plane	affine, plane	a, p
radically integral local field	radically, integral, local, field	r, i, l, f
open oriented line section	open, oriented, line, section	o, o, l, s
jacobian	jacobian	j
orthogonal kernel embedding	orthogonal, kernel, embedding	o, k, e

Concatenation: $a \cdot p \cdot r \cdot i \cdot l \cdot f \cdot o \cdot o \cdot l \cdot s \cdot j \cdot o \cdot k \cdot e = \texttt{aprilfoolsjoke}$ . $\square$

Editorial

We enumerate the admissible parameter range, discard candidates that violate the derived bounds or arithmetic constraints, and update the final set or total whenever a candidate passes the acceptance test.

Pseudocode

    bold_words = ["affine", "plane", "radically", "integral", "local",
                  "field", "open", "oriented", "line", "section",
                  "jacobian", "orthogonal", "kernel", "embedding"]
    Return concatenate(w[0] for w in bold_words)

Complexity Analysis

Time: $O(1)$ .
Space: $O(1)$ .

Answer

\boxed{\texttt{aprilfoolsjoke}}

C++ project_euler/problem_836/solution.cpp

#include <bits/stdc++.h>
using namespace std;

int main() {
    // Problem 836: A Bold Proposition (April Fools' 2023)
    // The answer is the concatenation of the first letters of each bolded word:
    // affine plane, radically integral local field,
    // open oriented line section, jacobian,
    // orthogonal kernel embedding
    // => a p r i l f o o l s j o k e => aprilfoolsjoke

    cout << "aprilfoolsjoke" << endl;
    return 0;
}

Python project_euler/problem_836/solution.py

# Problem 836: A Bold Proposition (April Fools' 2023)
# The answer is the concatenation of the first letters of each bolded word:
# affine plane, radically integral local field,
# open oriented line section, jacobian,
# orthogonal kernel embedding
# => a p r i l f o o l s j o k e => aprilfoolsjoke

bold_words = [
    "affine", "plane",
    "radically", "integral", "local", "field",
    "open", "oriented", "line", "section",
    "jacobian",
    "orthogonal", "kernel", "embedding"
]

answer = "".join(w[0] for w in bold_words)
print(answer)