The nine cells of the triangle contain nine zeros. Try to change the numbers, with any number of moves, so that they become nine consecutive positive integers, the smallest of which is a. The arrangement of these nine numbers in the triangle is irrelevant. A move is the increase or decrease by 1 of the two numbers occupying two adjacent cells, where adjacent cells share a side. For which values of a is the problem solvable?
A pair of adjacent cells always consists of one orange and one black triangle. Initially, the sum r of the numbers on the orange cells is equal to the sum b of the numbers on the black cells, namely, 0. A move does not change this equilibrium r = b. At the end, the nine numbers of the triangle should be a , a + 1, …, and a + 8. Thus, because b is the sum of six of the numbers, b ≥ a + (a + 1) + (a + 2) + (a + 3) + (a + 4) + (a + 5) = 6a + 15, and because r is the sum of three of the numbers, r ≤ (a + 6) + (a + 7) + (a + 8) = 3a + 21. Because b = r, 6a + 15 ≤ 3a + 21, or a ≤ 2. Furthermore, r = b means that r + b = 2r, so r + b must be an even number. For a = 1, the sum of the nine numbers 1 + 2 + … + 9 = 45, which is odd. Therefore, if there is a solution to the problem at all, it is only for a = 2.
Instead of increasing the numbers on the same pair of cells by 1 n times, one can directly increase both numbers by n. This does not change the problem in any way. The order of the individual moves also plays no role. The image shows in three steps that there is indeed a solution for a = 2.
On supporting science journalism
If you’re enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.
We’d love to hear from you! E-mail us at games@sciam.com to share your experience.
This puzzle originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.
Subscribe to Support Independent Journalism
Great science journalism requires human expertise, time, effort and creativity. And it costs money. That’s why I and the journalists here at Scientific American hope you’ll join our community.
When you subscribe, you are supporting staff and freelance journalists who are passionate about telling science stories that are true, important and compelling. Our editors and reporters are often experts in their fields, which means they understand the nuances of big discoveries and can untangle the breakthroughs from the hype. With a subscription, you are also supporting rigorous fact-checking to ensure the words we publish are precise and accurate. And you’re supporting original illustrations, graphics and photos that bring you closer to an advanced laboratory, an ice sheet in Antarctica or a space mission in orbit. You’re helping us craft other types of high-quality journalism as well: Our newsletters are carefully written, edited and curated by staffers you have or will come to know and love. Our Science Quickly podcast is based on original reporting, collaboration with editors and scientists and exacting production.
Subscriptions keep this engine running so we can continue to deliver thoughtful, rigorous and independent science journalism to you. In an era of viral misinformation, this work is crucial. If you value what we do, I hope you’ll consider joining us as a subscriber