## 'Magic square' math puzzle has gone unsolved since 1996

Think you can solve it? One mathematician has already offered about $1,000 and a bottle of champagne to whoever cracks it first.

- The puzzle involves a particularly complicated type of magic square.
- Magic squares are square arrays containing distinct numbers, and the sums of the numbers in the columns, rows and diagonals must be equal.
- In 1996, the recreational mathematics writer Martin Gardner offered $100 to whoever could solve a 3x3 magic square — but using squared numbers.

docdroid.net

<p>Given that you need each column, row and diagonal to add up to 15, you'd need to fill in the empty squares with a 9, 7 and 8. </p>docdroid.net

<p>That may be easy enough. But magic squares become far more difficult when they use squared numbers, a concept <a href="https://www.scientificamerican.com/article/can-you-solve-a-puzzle-unsolved-since-1996/" target="_blank">first exemplified</a> by the 18th-century mathematician Leonhard Euler. </p><p>Since, mathematicians have generated various configurations of 4x4 magic squares of squares, including 5x5, 6x6 and 7x7 versions. But nobody has yet proven that a 3x3 magic square of squares is possible — or impossible, for that matter.</p><p>To date, there have been at least two prizes offered to whoever can solve this longstanding puzzle. Martin Gardner, a science and mathematics writer who was perhaps best known for devising recreational mathematics games that appeared for 25 years in a column published by <em>Scientific American,</em> offered a prize of $100 in 1996 to whoever could crack the code first. </p><ul> <p>"So far no one has come forward with a "square of squares"—but no one has proved its impossibility either," Gardner wrote in 1998 in <em><a href="https://blogs.scientificamerican.com/observations/a-quarter-century-of-recreational-m-2010-05-26/" target="_blank">Scientific American</a></em>. "If it exists, its numbers would be huge, perhaps beyond the reach of today's fastest supercomputers."</p></ul>*Melancholia I. (A 4x4 magic square is depicted in the top right of the painting.)*

Dürer's

<p>In 2005, the mathematician Christian Boyer raised the stakes by offering €1,000 plus a bottle of champagne to anyone who could complete a 3x3 magic square of squares — using seven, eight or nine distinct squared integers. (Boyer also offered a prize for anyone who can show the puzzle is impossible, and he lists smaller prizes for other unsolved puzzles on his <a href="http://multimagie.com/indexengl.htm" target="_blank">website</a>.)</p><p>While both prizes remain unclaimed, some people have come close to solving the 3x3 magic square of squares, like this configuration listed on Christian Boyer's website.</p>## Can you solve what an MIT professor once called 'the hardest logic puzzle ever'?

Logic puzzles can teach reasoning in a fun way that doesn't feel like work.

- Logician Raymond Smullyan devised tons of logic puzzles, but one was declared by another philosopher to be the hardest of all time.
- The problem, also known as the Three Gods Problem, is solvable, even if it doesn't seem to be.
- It depends on using complex questions to assure that any answer given is useful.

### The Three Gods Problem

<iframe width="730" height="430" src="https://www.youtube.com/embed/UyOGZk7WbIk" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe><p> One of the more popular wordings of the problem, which MIT logic professor George Boolos <a href="https://www.readersdigest.ca/culture/hardest-logic-puzzle-ever/" target="_blank">said</a> was the hardest ever, is:<br> <br> "Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for <em>yes</em> and <em>no</em> are <em>da</em> and <em>ja</em>, in some order. You do not know which word means which."<br> <br> Boolos adds that you are allowed to ask a particular god more than one question and that Random switches between answering as if they are a truth-teller or a liar, not merely between answering "da" and "ja." <br> <br> Give yourself a minute to ponder this; we'll look at a few answers below. Ready? Okay. <strong><br> <br></strong>George Boolos' <a href="https://www.pdcnet.org/8525737F00588A37/file/31B21D0580E8B125852577CA0060ABC9/$FILE/harvardreview_1996_0006_0001_0060_0063.pdf" target="_blank" rel="noopener noreferrer">solution</a> focuses on finding either True or False through complex questions. </p><p> In logic, there is a commonly used function often written as "iff," which means "if, and only if." It would be used to say something like "The sky is blue if and only if Des Moines is in Iowa." It is a powerful tool, as it gives a true statement only when both of its components are true or both are false. If one is true and the other is false, you have a false statement. </p><p> So, if you make a statement such as "the moon is made of Gorgonzola if, and only if, Rome is in Russia," then you have made a true statement, as both parts of it are false. The statement "The moon has no air if, and only if, Rome is in Italy," is also true, as both parts of it are true. However, "The moon is made of Gorgonzola if, and only if, Albany is the capitol of New York," is false, because one of the parts of that statement is true, and the other part is not (The fact that these items don't rely on each other is immaterial for now).</p><p> In this puzzle, iff can be used here to control for the unknown value of "da" and "ja." As the answers we get can be compared with what we know they would be if the parts of our question are all true, all false, or if they differ. </p><p> Boolos would have us begin by asking god A, "Does "da" mean yes if and only if you are True if and only if B is Random?" No matter what A says, the answer you get is extremely useful. As he explains: <br> </p><p> "If A is True or False and you get the answer da, then as we have seen, B is Random, and therefore C is either True or False; but if A is True or False and you get the answer ja, then B is not Random, therefore B is either True or False… if A is Random and you get the answer da, C is not Random (neither is B, but that's irrelevant), and therefore C is either True or False; and if A is Random...and you get the answer ja, B is not random (neither is C, irrelevantly), and therefore B is either True or False."<br> <br> No matter which god A is, an answer of "da" assures that C isn't Random, and a response of "ja" means the same for B. </p><p> From here, it is a simple matter of asking whichever one you know isn't Random questions to determine if they are telling the truth, and then one on who the last god is. Boolos suggests starting with "Does da mean yes if, and only if, Rome is in Italy?" Since one part of this is accurate, we know that True will say "da," and False will say "ja," if faced with this question. </p><p> After that, you can ask the same god something like, "Does da mean yes if, and only if, A is Random?" and know exactly who is who by how they answer and the process of elimination. </p><p> If you're confused about how this works, try going over it again slowly. Remember that the essential parts are knowing what the answer will be if two positives or two negatives always come out as a positive and that two of the gods can be relied on to act consistently. </p><p> Smullyan wrote several books with other logic puzzles in them. If you liked this one and would like to learn more about the philosophical issues they investigate, or perhaps if you'd like to try a few that are a little easier to solve, you should consider reading them. A few of his puzzles can be found with explanations in this <a href="https://www.nytimes.com/interactive/2017/02/11/obituaries/smullyan-logic-puzzles.html" target="_blank" rel="noopener noreferrer">interactive</a>. </p>## Paradox-free time travel is 'logically' possible, say physicists

Grandfathers, take heart. You'll survive the paradox that's been gunning for you since the 1930s.

### A paradox primer

<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yNDQ1MzcyOC9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTYwNjgyOTE3Mn0.sdkQ7EF5R2l1zED3BzbpcNG6FFy5Rf1OjqS9u5RC1JU/img.jpg?width=1245&coordinates=0%2C75%2C0%2C75&height=700" id="03234" class="rm-shortcode" data-rm-shortcode-id="9ca569af0bbe83100698d67202e4bcbf" data-rm-shortcode-name="rebelmouse-image" />According to the study, the universe would have worked things out whether Marty stole credit for "Johnny B. Goode" or not.

(Photo: Universal Studios)

<p>The classic temporal thought experiment is known as <a href="https://www.space.com/grandfather-paradox.html#:~:text=The%20grandfather%20paradox%20is%20a%20potential%20logical%20problem%20that%20would,make%20their%20own%20birth%20impossible." target="_blank">the grandfather paradox</a>. It goes like this: Imagine you decide to go back in time to kill your grandfather. Yes, his election-year posts have been that embarrassing. You travel back and kill him before he conceives one-half of your parents. But then, how is it you can exist to go back and kill him? But if you don't exist, then who killed your grandfather? <a href="https://bigthink.com/paul-ratner/neil-degrasse-tyson-explains-the-strange-paradoxes-of-time-travel" target="_self">Paradox</a>. The timeline is no longer self-consistent. (<a href="https://www.youtube.com/watch?v=XayNKY944lY" target="_blank">Maybe</a>.)</p><p>You can play this game with most time traveling tales. In "<a href="https://www.imdb.com/title/tt0088763/" target="_blank" style="">Back to the Future</a>," Marty travels back in time and interferes with his parents' dalliance, preventing himself from being born. But if Marty is never born, how does he interfere with his parents' dalliance? But if he can't interfere, what's preventing him from being born? And round we go.</p><p>One would think such worries limited to high-minded philosophy debates or low-brow movie riffs. But some solutions to Einstein's field equations allow time travel through <a href="https://www.thegreatcoursesdaily.com/did-einstein-prematurely-reject-godels-universe/#:~:text=A%20closed%20timelike%20curve%20is,encounter%20the%20same%20event%20again." target="_blank" rel="noopener noreferrer">closed timelike curves</a>. These theoretical paths would allow someone to be present at an initial event, travel through space and time, and return to that event again. Think a spacetime loop-the-loop. Importantly, the return point is not a repeat of the initial event. It is the initial event.</p><p>The implications of closed timelike curves lead to all sorts of wild time travel scenarios. <a href="https://bigthink.com/dr-kakus-universe/is-time-travel-possible" target="_self">According to physicist Michio Kaku</a>, these have included traveling through a wormhole, through a spinning black hole, around an infinitely-long spinning cylinder, and around two colliding cosmic strings.</p>### The universe is a self-regulating Time Lord

<img type="lazy-image" data-runner-src="https://assets.rebelmouse.io/eyJhbGciOiJIUzI1NiIsInR5cCI6IkpXVCJ9.eyJpbWFnZSI6Imh0dHBzOi8vYXNzZXRzLnJibC5tcy8yNDQ1MzczNC9vcmlnaW4uanBnIiwiZXhwaXJlc19hdCI6MTYzMTQ2NTA3Mn0.QawOiC0smajTijNpoJbY1UsnB4VhoRGds5swcKdowW8/img.jpg?width=1245&coordinates=0%2C51%2C0%2C51&height=700" id="d0230" class="rm-shortcode" data-rm-shortcode-id="93e944fc3f902cadf33e6e7211efc84d" data-rm-shortcode-name="rebelmouse-image" />Dr. Fabio Costa (left) and Germain Tobar (right) discuss their findings. Behind them, a process function (*w*) interacts with localized spacetime regions with closed timelike curves.

Credit: University of Queensland

<p>With time travel on the theoretical table, Tobar Germain, a University of Queensland undergraduate, wanted to test its consistency. Is paradox-free time travel mathematically possible? To answer that question, he teamed up with Dr. Fabio Costa, a University of Queensland physicist, to crunch the numbers.</p><p>"Some physicists say it is possible, but logically, it's hard to accept because that would affect our freedom to make any arbitrary action," Tobar said <a href="https://www.uq.edu.au/news/article/2020/09/young-physicist-squares-numbers%E2%80%99-time-travel" target="_blank">in a release</a>. "It would mean you can time travel, but you cannot do anything that would cause a paradox to occur."</p><p>According to their research, time travel can be consistent and free of logical paradoxes. However, that requires the outputs of all but two space-time regions to be fixed. In that case, despite the presence of closed timelike loops, entities can maintain their freedom of choice without resulting in a paradox.</p><p>"The maths checks out, and the results are the stuff of science fiction," Costa said in the same release.</p><p>To illustrate their findings, Tobar and Costa offer a thought experiment straight out of science fiction. Imagine you travel through time to stop the COVID-19 pandemic. You locate and quarantine patient zero. Mission (and paradox) accomplished, right? Not according to their research. The math suggests that temporal events would adjust to being logically consistent with any action you made. For example, you may catch the virus, become patient zero, and spread the pandemic anyway. </p><p>Therefore, future, erm, past you still has the stimulus that sent you back in time initially.</p><p>"No matter what you did, the salient events would just recalibrate around you," Tobar said. "That would mean that—no matter your actions—the pandemic would occur, giving your younger self the motivation to go back and stop it.</p><p>"The range of mathematical processes we discovered show that time travel with free will is logically possible in our universe without any paradox."</p>### Riding the timelike curve?

<span style="display:block;position:relative;padding-top:56.25%;" class="rm-shortcode" data-rm-shortcode-id="26aeff2dbb93f6414170073a6f60c870"><iframe type="lazy-iframe" data-runner-src="https://www.youtube.com/embed/6yMiUq7W_xI?rel=0" width="100%" height="auto" frameborder="0" scrolling="no" style="position:absolute;top:0;left:0;width:100%;height:100%;"></iframe></span><p>Of course, sayings paradox-free time travel is mathematically consistent is a wildly different statement than saying it is practically possible. Even if you could take the plunge into <a href="https://www.scientificamerican.com/article/the-chronology-protection/" target="_blank" rel="noopener noreferrer">a wormhole</a>, there's a good chance you'd be crushed out of existence before reaching the other end. Souped-up DeLorean or no. </p><p>It all depends on how the <a href="https://bigthink.com/surprising-science/physicist-radical-theory-of-gravity" target="_self">laws of quantum gravity</a> shake out, and physicists are still exploring that very open question. What about those other scenarios Kaku pointed out? In <a href="https://bigthink.com/dr-kakus-universe/is-time-travel-possible-part-ii" target="_self">a follow-up article</a>, he points out that none can be realized using known physical mechanisms.</p><p>So, while we may be the time lords of the whiteboard, the universe will be a one-way street for the foreseeable future.</p>## How math predicts life on Earth and the universe beyond

Math doesn't suck. It is one of humanity's greatest and most mysterious journeys.

- There is a pervasive cultural attitude against mathematics, but it is actually a mind-blowing tool for analyzing and predicting the world around us—and far beyond. We asked mathematicians Edward Frenkel and Po-Shen Loh, and physicists Michio Kaku, Michelle Thaller, Janna Levin and Geoffrey West to explain the wonders of math.
- West explains the rule of 'quarter-power scaling' in biology—there is a mathematical equation that predicts how much food an organism needs to eat to survive and it's remarkably consistent, whether you're looking at ladybugs, cats, elephants, and even trees and flowers. Math underpins our lives in incredible ways.
- Infinitesimal calculus—the math that describes how moving bodies change over time—turns out to predict not just phenomena on Earth but far out in the universe. The 11-dimensional math used by physicists turns out to predict the exact results of particle physics experiments. Humanity is on an incredible journey with mathematics and every day it opens up the world and universe in eye-opening ways.

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