An entry in Fortnightly Topic Challenge #38: Reusing Information 1

A Bongard Problem is a problem where you have 6 symbols on either side of a line. The symbols on the left follow a certain rule, while the symbols on the right do not follow it. The objective is to find this rule.

I have made 5 Bongard problems for the community. I have enumerated the symbols so that it will be easier to reference/discuss them. The numbers themselves are not part of the symbols and should be ignored when applying the rule in question.

  1. enter image description here

  2. enter image description here

  3. enter image description here

  4. enter image description here

  5. enter image description here

Now tell me, what are the six rules?


Rule 5: The number of ____ is one more than a multiple of 3.

  • 1
    $\begingroup$ Wait! I got it! For the first one the ones on the left side all have the numbers are multiples of 4 plus 1 and and multiples of 4 plus 2! $\endgroup$ – Yout Ried Sep 15 '18 at 16:48
  • $\begingroup$ Is that a skechters symbol I see? $\endgroup$ – Yout Ried Sep 15 '18 at 16:52
  • $\begingroup$ @YoutRied Nice try but the numbers aren't part of the rule $\endgroup$ – Riley Sep 15 '18 at 16:55
  • $\begingroup$ it was a joke :P Nice puzzle though. I might try to answer it later but i gotta do homework now :P $\endgroup$ – Yout Ried Sep 15 '18 at 16:55
  • $\begingroup$ Well, symbol $35$ denotes water, doesn't it? $22$ is the famous "S" that every kid knows of. $41$ is the peace symbol. $56$ is just a baseball or tennis ball (but since there are "stitches" in it, I am guessing it's the former.) $\endgroup$ – user477343 Sep 21 '18 at 6:15

Partial answer


If you were to colour each region in an image (including the outer region) so that no two adjacent regions have the same colour (see Four Colour Theorem), the images on the right would all need at least three colours while those on the left can all be coloured with less than three. In particular, the images on the right all have a set of three regions which are mutually adjacent to one another while those on the left do not.


The number of intersection points of degree $3$ (points where two or more distinct curves meet) is one more than a multiple of three for the images on the left. Here is the highlighted image to show what I mean enter image description here

  • $\begingroup$ Yes, this is correct. The rule is rot13(gjb-pbybenovyvgl). $\endgroup$ – Riley Apr 12 at 13:46
  • $\begingroup$ #5 is correct. The hint probably made it pretty easy, but it would've been too hard to find without it. $\endgroup$ – Riley Apr 12 at 14:48

A possible rule for the first problem:

In the first Bongard problem, count the number of segments in each shape. Segments are the lines between any sort of conjunction. On the left the count is a prime or a power of two. Here's an image: https://m.imgur.com/TDQ38RS

A possible rule for the second problem:

Count the number of tails protruding from the largest loop. Odds on the right, evens on the left. Here's an image showing the loops and tails:https://m.imgur.com/gTbnoQU

Deleted my first version of Rule 2 because too ambiguous what continue straight meant

A possible rule for number 5:

Connect all possible dots, then count the number of sections in the shape. Multiples of 3 are on the left. Here's an image:https://m.imgur.com/4NKZREN

  • $\begingroup$ Unfortunately, neither is correct. I'm not entirely sure how you define "segment" or "stroke," but the intended rules are simpler. $\endgroup$ – Riley Apr 12 at 0:09
  • $\begingroup$ I added another possibility. Probably still more complicated than what you're looking for $\endgroup$ – Artemmm Apr 12 at 1:07
  • $\begingroup$ Your update for the second rule is so close to what I intended, and since I didn't add any counterexamples, I'll accept it. The intended rule was rot13(Nyy raqcbvagf pna or cnverq bss jvgubhg pebffvat nal yvarf). imgur.com/YHgcLPO $\endgroup$ – Riley Apr 12 at 1:29
  • $\begingroup$ @Riley I didn't think of the second bit so #16 broke it for me. Thats why I only counted the outs $\endgroup$ – Artemmm Apr 12 at 2:02
  • $\begingroup$ #5 is not the intended rule, although it's weird how it works out. I think my rule #5 was unnecessarily complicated, so I will give a hint or two. $\endgroup$ – Riley Apr 12 at 13:52

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