# Math problems that sound hard, but are actually easy **URL:** \ **Category:** Miscellaneous and Personal Stuff I Must Share\ **Created:** [January 23, 2026, 7:14pm UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965 "2026-01-23T19:14:09Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Timz](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Timz](https://boards.straightdope.com/u/Timz)\ **Post date:** [January 23, 2026, 7:14pm UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/1 "2026-01-23T19:14:09Z")
Multiply together all the prime numbers from 2 up thru n, then add 1 to the result. Show that the final total is never a perfect square. No doubt lots of you know how easy that one is. So, what’s a better example of a problem that stumps people in spite of being easy? If you’re not already acquainted with the above example, being told that it’s easy helps you solve it. Can you find a problem where hearing that it’s easy doesn’t help? ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [January 23, 2026, 7:31pm UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/2 "2026-01-23T19:31:31Z")
This isn’t true. For example for N=4, we have 2 times 3 times 4 plus 1 is 25, which is a perfect square. It isn’t a perfect square of any integer up to N, which may be the proof you were thinking of ETA: **Erroneous** answer now spoiler protected, so I don’t confuse other people. ---
**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\ **Post date:** [January 23, 2026, 7:33pm UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/3 "2026-01-23T19:33:26Z")
> [@Andy\_L](#): > > For example for N=4, we have 2 times 3 times 4 plus 1 is 25, which is a perfect square. **Primes** up to N. For N=4, we have 2\*3+1 = 7, which is not a perfect square ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [January 23, 2026, 7:39pm UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/4 "2026-01-23T19:39:47Z")
D’oh. My apologies. I misread the OP and now must slink off in shame. ---
**Author:** ![Timz](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Timz](https://boards.straightdope.com/u/Timz)\ **Post date:** [January 23, 2026, 8:08pm UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/5 "2026-01-23T20:08:16Z")
This one is plainly easy, as long as you’re 75+ years old. How easy is it for younger people? (Pencil and paper allowed, nothing else.) How many digits in 5 to the 300th power? ---
**Author:** ![xtenkfarpl](https://avatars.discourse-cdn.com/v4/letter/x/c2a13f/32.png) [@xtenkfarpl](https://boards.straightdope.com/u/xtenkfarpl)\ **Post date:** [January 23, 2026, 9:12pm UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/6 "2026-01-23T21:12:15Z")
A converse is Goldbachs’s conjecture. Every even number can be expressed as the sum ot two primes. Easy to state and just about anyone can understand it. But still not proved. ---
**Author:** ![Q.Q.Switcheroo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/q.q.switcheroo/32/5713_2.png) [@Q.Q.Switcheroo](https://boards.straightdope.com/u/Q.Q.Switcheroo)\ **Post date:** [January 23, 2026, 11:09pm UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/7 "2026-01-23T23:09:41Z")
In 30 seconds or less, add all the whole numbers from 1 to 100. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 23, 2026, 11:10pm UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/8 "2026-01-23T23:10:13Z")
> [@Timz](#): > > This one is plainly easy, as long as you’re 75+ years old. The only thing I can think of is that people that old would have worked with logarithm tables enough that they’d have memorized the common log of 5? ---
**Author:** ![BobLibDem](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/boblibdem/32/3149_2.png) [@BobLibDem](https://boards.straightdope.com/u/BobLibDem)\ **Post date:** [January 23, 2026, 11:13pm UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/9 "2026-01-23T23:13:04Z")
That’s easy. You have 1+100, 2+99, 3+98… Fifty pairs of numbers each adding up to 101. 50 x 101 = 5050. ---
**Author:** ![Timz](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Timz](https://boards.straightdope.com/u/Timz)\ **Post date:** [January 23, 2026, 11:25pm UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/10 "2026-01-23T23:25:52Z")
> [@Thudlow\_Boink](#): > > people that old would have worked with logarithm tables enough that they’d have memorized the common log of 5 Just about right. If you used logarithms, you remember the base-10 log of 2 is a musical 0.30103, so the log of 5^300 is 300 minus (300 x 0.30103). ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 23, 2026, 11:26pm UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/11 "2026-01-23T23:26:00Z")
> [@Great\_Antibob](#): > > **Primes** up to N. In @Andy_L’s defense, n is a whole number; p is a prime. ---
**Author:** ![Andy\_L](https://avatars.discourse-cdn.com/v4/letter/a/c67d28/32.png) [@Andy\_L](https://boards.straightdope.com/u/Andy_L)\ **Post date:** [January 24, 2026, 12:53am UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/12 "2026-01-24T00:53:25Z")
In expiation, here’s one. I’m not telling you what base I’m working in - can you if 1001 is a prime number, even with that handicap? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [January 24, 2026, 1:13am UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/13 "2026-01-24T01:13:54Z")
That’s x^3 + 1, where x is the base. Which factors to (x+1)(x^2-x+1). So whatever your base is, 1 plus that is a factor.. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [January 24, 2026, 1:27am UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/14 "2026-01-24T01:27:11Z")
OK, here’s an old favorite of mine: Simplify (x-a)\*(x-b)\*(x-c)\*...\*(x-z) ---
**Author:** ![markn\_1](https://avatars.discourse-cdn.com/v4/letter/m/f9ae1b/32.png) [@markn\_1](https://boards.straightdope.com/u/markn_1)\ **Post date:** [January 24, 2026, 2:00am UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/15 "2026-01-24T02:00:07Z")
Heh. That’s zero, because of the (x-x) factor. ---
**Author:** ![beowulff](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/beowulff/32/542_2.png) [@beowulff](https://boards.straightdope.com/u/beowulff)\ **Post date:** [January 24, 2026, 2:16am UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/16 "2026-01-24T02:16:57Z")
It’s easy if you are Gauss. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 24, 2026, 2:49am UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/17 "2026-01-24T02:49:25Z")
> [@Timz](#): > > Multiply together all the prime numbers from 2 up thru n, then add 1 to the result. Show that the final total is never a perfect square. No one has yet answered this, so I’ll give my answer. I don’t remember ever seeing this problem before, and the answer wasn’t immediately obvious, but it is relatively straightforward. I’m curious whether my solution is the intended “easy” one. Let P represent the product of “all the prime numbers from 2 up thru n.” Suppose P + 1 = k² for some whole number k. Since P has a factor of 2, P is even, and so P+1 = k² and k are odd. P = k² – 1 = (k – 1)(k + 1), and since k is odd, both k–1 and k+1 must be even. But P has only one factor of 2, so we have a contradiction. ---
**Author:** ![BigT](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bigt/32/12044_2.png) [@BigT](https://boards.straightdope.com/u/BigT)\ **Post date:** [January 24, 2026, 2:50am UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/18 "2026-01-24T02:50:34Z")
> [@Chronos](#): > > That’s x^3 + 1, where x is the base. Which factors to (x+1)(x^2-x+1). So whatever your base is, 1 plus that is a factor.. and thus 1001 in any base is always divisible by 11 ---
**Author:** ![BobLibDem](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/boblibdem/32/3149_2.png) [@BobLibDem](https://boards.straightdope.com/u/BobLibDem)\ **Post date:** [January 24, 2026, 3:22am UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/19 "2026-01-24T03:22:40Z")
Yeah that’s how the famous story went. ---
**Author:** ![Timz](https://avatars.discourse-cdn.com/v4/letter/t/6a8cbe/32.png) [@Timz](https://boards.straightdope.com/u/Timz)\ **Post date:** [January 24, 2026, 7:44pm UTC](https://boards.straightdope.com/t/math-problems-that-sound-hard-but-are-actually-easy/1026965/20 "2026-01-24T19:44:55Z")
Another answer to the multiply-primes problem is similar to Thudlow\_Boink’s. You multiply 2 and a string of odd numbers, so the product is an even number but not a multiple of 4 – so it’s 2 more than a multiple of four, and the final total is 3 more than a multiple of 4, which a square never is. (The square of an odd number 2n + 1 is 4n^2 + 4n + 1.)