{"id":935,"date":"2017-09-26T16:36:17","date_gmt":"2017-09-26T20:36:17","guid":{"rendered":"http:\/\/mathriddles.williams.edu\/?p=935"},"modified":"2025-01-28T13:04:31","modified_gmt":"2025-01-28T18:04:31","slug":"sum-things-are-products-of-twisted-minds","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/mathriddles\/restricted\/sum-things-are-products-of-twisted-minds\/","title":{"rendered":"Sum Things are Products of Twisted Minds"},"content":{"rendered":"

Sum Things are Products of Twisted Minds<\/strong><\/p>\n

First Thoughts:<\/strong><\/p>\n

It may seem to the uninitiated observer that no concrete information was communicated between S and P in their discussion, but if we assume that both are completely rational agents, we can in fact figure out the numbers x and y. Each statement that S or P makes communicates vital information to the reader. We thus decide to proceed by examining them one at a time to decode their content. Before we even get this far, however, we may note that since x and y are both at least 2, we know that their sum and product are both at least 4. Furthermore, the product xy cannot be a prime number, since neither x nor y can be equal to 1.<\/p>\n

 <\/p>\n

Second Thoughts:<\/strong><\/p>\n

When S announces that he does not know what the numbers are, we know that the sum x + y must be realizable in at least two different ways as a sum of integers greater than or equal to 2. This eliminates x + y = 4 and x + y = 5, because each of these can only be expressed as such a some in a single way, up to order. These sums are 2 + 2 = 4 and 2 + 3 = 5 respectively. When we consider x + y = 6, we see that we could either have 4 + 2 = 6 or 3 + 3 = 6. Any larger sums will also be expressible in more than one way. Therefore, we have discovered that x + y is at least 6. This seems to be all we can glean from S at the moment.<\/p>\n

 <\/p>\n

Third Thoughts:<\/strong><\/p>\n

Next, we move on to P\u2019s statement. He reveals two pieces of information: he knows that S did not know the two numbers, and he does not know the product. For the first of these, P is saying in effect that he knows that the sum x + y is at least 6, since he is able to follow the same chain of logic as S. Therefore, he knows that x and y are not equal to 2 and 2 or 2 and 3. It follows that our product is at least 6.<\/p>\n

 <\/p>\n

Examining the second statement, we see that if x and y are both prime numbers, then P would know the product. Therefore, it must be the case that at most one of x and y is prime. What does this mean for the product xy? Since at most one of x and y is prime, the product xy must have at least 3 prime factors, although these factors may not necessarily be unique. For example, we could have xy = 16 = 2 * 2 * 2 * 2 or xy = 12 = 2 * 3 * 3. Furthermore, we must also have that xy is not the cube of a prime. If xy = p^3, then the only possible decomposition, up to order, is x = p and y = p^2.<\/p>\n

 <\/p>\n

Final Thoughts:<\/strong><\/p>\n

We have come to the final exchange of information between the two interlocutors. S is able to ascertain from P\u2019s clues the identity of the two numbers x and y. This might initially give us pause, because we are not able to make a similar discovery, but our situation is a result of us having neither knowledge of the sum nor product. S is more privileged than us, because he already knows the sum. Now we have to make sense of how S is able to figure out the numbers.<\/p>\n

We already know that the sum S knows can be expressible in more than one way. If all of these decompositions of the sum resulted in both x and y being prime, then P would know the product. It follows that there is at least one decomposition where either x or y is not prime. If there was more than one decomposition where this was the case, then S would not be able to determine the numbers. Therefore, there is exactly one decomposition where x or y is not prime. All other decompositions must consist of prime numbers added together.<\/p>\n

Let\u2019s use this information to find an upper bound on our sum. The first two composite numbers larger than 2 are 4 and 6. Therefore, any integer that can be decomposed in two separate ways, one of which involves a 4 and one of which involves a 6, is too large to be our sum. Since x and y are both at least 2, we see that the smallest such integer is 2 + 6 = 8 = 4 + 4. Any larger integer n can be written as n = (n \u2013 4) + 4 = (n \u2013 6) + 6. Therefore, our sum is at most 7.<\/p>\n

We only have two candidates for the sum to consider: 6 and 7. We see that there are two possible decompositions for each: 6 = 2 + 4 = 3 + 3 and 7 = 2 + 5 = 3 + 4. We consider them in this order. For the decompositions of 6, we see that 2 * 4 = 8 = 2^3 and 3 * 3 = 9. Neither of these products meet our established criteria for products, so we throw them out. This leaves 7. Since 2 and 5 are both prime, we must have x and y are 3 and 4, resulting in xy = 12.<\/p>\n","protected":false},"excerpt":{"rendered":"

Sum Things are Products of Twisted Minds First Thoughts: It may seem to the uninitiated observer that no concrete information was communicated between S and P in their discussion, but if we assume that both are completely rational agents, we can in fact figure out the numbers x and y. Each statement that S or…<\/p>\n","protected":false},"author":2861,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[27],"tags":[],"class_list":["post-935","post","type-post","status-publish","format-standard","hentry","category-restricted"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/mathriddles\/wp-json\/wp\/v2\/posts\/935","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/mathriddles\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.williams.edu\/mathriddles\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/mathriddles\/wp-json\/wp\/v2\/users\/2861"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/mathriddles\/wp-json\/wp\/v2\/comments?post=935"}],"version-history":[{"count":1,"href":"https:\/\/sites.williams.edu\/mathriddles\/wp-json\/wp\/v2\/posts\/935\/revisions"}],"predecessor-version":[{"id":1118,"href":"https:\/\/sites.williams.edu\/mathriddles\/wp-json\/wp\/v2\/posts\/935\/revisions\/1118"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/mathriddles\/wp-json\/wp\/v2\/media?parent=935"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/mathriddles\/wp-json\/wp\/v2\/categories?post=935"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/mathriddles\/wp-json\/wp\/v2\/tags?post=935"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}