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Numberphile solves the square-sum problem by reversing the sequence, then multiplying by a square number, and then applying the shift function. This technique can be used to solve any sum sequence that is regular.

**00:00:00**The Numberphile discusses a recently solved problem in mathematics known as the square-sum problem. The problem asks whether there exists a sequence of numbers from 1 to n such that the sum of any two consecutive numbers is a square. The video introduces the mystery problem by asking a question about arranging the numbers from 1 to 15 in such a way that the sum of any two consecutive numbers is a square. The problem breaks at 18, and adding 19 or 20 does not produce a solution. The Numberphile then introduces the first critical idea: looking for regular sequences of numbers that we already found. The Numberphile hands out a regular sequence of numbers, and asks the viewer to find a new, longer regular sequence without having to trial and error. The viewer has exactly 5 seconds to pause if they want to try. After the viewer pauses, the Numberphile starts with the first number in the sequence.**00:05:00**Numberphile solves the square-sum problem by reversing the sequence, then multiplying by a square number, and then applying the shift function. This technique can be used to solve any sum sequence that is regular.**00:10:00**The Numberphile solves the square-sum problem in SoME2. The problem is that numbers can appear more than once when multiplied by 4. By multiplying the sequence by 9 and then shifting it by 4, they are able to create new sequences that do not have any duplicates.**00:15:00**Numberphile demonstrates how to solve the square sum problem for arbitrary numbers using a simple recipe, and also provides a definition for a ninja pair: two sequences that are both regular, and have the same first and last number, but differ by an integer increment.**00:20:00**Numberphile solves the square-sum problem for small numbers by constructing new ninja pairs from existing pairs. This process goes on until every number from 99 to 4,900 has a solution.**00:25:00**Numberphile explains how they solved the square-sum problem in SoME2, which was famously unsolved by mathematicians. They show how reasoning helped them to understand what formulas they needed the computer to find.

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