Grid numbers

EL PAÍS and the Royal Spanish Mathematical Society present their first summer math challenge


In partnership with the Royal Spanish Mathematical Society, EL PAÍS presents its first math challenge of this summer. A new math problem will be published every Thursday until the end of August and readers will have until a minute before midnight on Monday night to send in their answers. Winners will be selected at random from the correct answers and announced on Tuesdays.

Winners will receive the book collection Grandes Ideas de la Ciencia (Great Science Ideas).

The first challenge is set by Adolfo Quirós, who teaches at Madrid’s Autónoma University and is also vice-president of the Royal Spanish Mathematical Society. As well as coming up with this challenge, he has coordinated this whole season of math problems.

Answers must be sent to before 00.00 on Tuesday, August 5 (midnight between Monday and Tuesday). In order to be correct, answers must provide an explanation of how the solution was reached, not just the solution itself.

To try to avoid any mistakes, we will now also explain the challenge in writing, as well as in the above video (in Spanish with English subtitles).

We begin with a four-by-four square grid in which each box contains either the value 1 or the value -1. The game involves changing the numbers around in some of the boxes, following a certain set of rules in order to end up with a 1 value inside every box.

There are two sets of rules to choose from:

- ACB rules (stricter): you may simultaneously change the values of every box in one entire row, one entire column, or one of the two diagonals.

-NBA rules (looser): besides the changes authorized by the ACB rules, you may also simultaneously change all the values in the boxes of an entire line running parallel to one of the two diagonals, which includes the possibility of just changing a corner box value.

We ask you to apply these rules to the two following grids:

For each grid, we ask whether it is possible to succeed in changing all the numbers to a 1 value using each set of rules, and if so, how to do it. The challenge is thus fourfold: for each grid, and using each set of rules, you must explain, in English or Spanish, the steps you used to complete the challenge – or explain why it is not possible to do so.

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