A 4×4 magic square with your chosen sum
Two constructions for a 4×4 magic square with any integer sum: the formula 21a+b and a transformation of the normal square with sum 34.
ReadMathematics
Articles by topic
Clear, organized learning paths that build strong foundations in mathematics and computer science.
Algebra
Fast mental calculation: recognize the structure before calculating
Eighteen explained techniques and their identities, 25 problems and an interactive practice laboratory.
ReadSix Kordemsky puzzles: with and without algebra
Six classic puzzles approached first through insight and then checked mathematically: coins, apples, averages, a current and two meetings.
ReadNotable sums: from Gauss to even and odd numbers
A chain of formulas born from one idea: pair the extremes, then transform the sum of natural numbers into sums of evens and odds.
ReadTen very challenging linear and quadratic word problems
Five reworked classics and five newly framed problems, with equations, constraints, rejected roots and step-by-step checks.
ReadThe same problem, different mathematics: from intuition to parameters
Eight situations explored in primary, lower-secondary, and upper-secondary terms: an advanced model helps the teacher design concrete explanations without imposing it on the child.
Read50 challenging linear word problems with solutions
Fifty challenging real-world problems solvable with first-degree equations. Solutions are hidden and, when useful, include a method without an equation.
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Number theory
Linear Diophantine equations: a general method using extended Euclid
Solve ax + by = c without guessing: theory and example side by side, Bézout’s identity, every integer solution and non-negativity constraints. With a step-by-step laboratory.
ReadThe hundred-animal problems: a general method and 20 worked solutions
A hundred animals, a hundred coins and three prices: Diophantine equations, gcd and congruences to find every solution, or prove that none exists. Twenty problems with click-to-reveal solutions.
ReadHourglasses: 12 problems, Diophantine equations and Bézout’s identity
A collection with explained solutions: from full cycles to partly full turns, from the GCD to impossible cases. Laboratory with extended Euclid and simulation.
ReadThe water-jug problem: from the book’s method to step-by-step pours
Averbach and Chein’s method: interpret integer coefficients as fills and empties, construct the pours and follow them in a laboratory with two animated bars.
ReadWho occupies the billionth position?
The exact solution to a sequence-position problem in which every natural number n appears exactly n times.
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