Logic Puzzles

Laura Givental took the lead on this activity for the 2012 Circle on the Road festival.

AUDIENCE: K- 3rd grade

If fruit is hidden in bags, but the labels on the bags are all wrong, can you identify which fruit is in which bag, by just looking in one bag? If there is a collection of hats, and several people put on some of the hats, can someone figure out the color of the hat on his or her own head based on the hats that are visible? It is fun to act out a number of similar scenerios and watch/guide children as they discover the logic. This is what we will do.


APPRENTICES: Rodi Steinig , In-Jae Kim, Marta Eso, Mark Saul, Sian Zelbo, Oleg Gleizer


This looks like fun. I was thinking that it would be fun to have some sort of narrative/drama that goes along with the game. (For instance, if we are doing the hat activity, why are they wearing hats in the first place? Maybe they are wizards and have a quest, or something like that.) Those kind of scenarios have worked really well in my math circles – the young kids really live it. What do you all think? (I’d be happy to come up with various scenarios for each game.)
BTW I have never led any logic games, and have done few. Could any of you point me in the direction of some good resources for (1) making sure I’m solid on the content and (2) learning a bit about their history. — Rodi

Here is a link to a nice collection of hat problems — we used some of these with our high school students in math team. Instead of hats, we got removable (post-it-like) labels and stuck these on the students’ backs.

Jim Tanton has a short video on youtube explaining the classic set-up:


We had classes on the described activities: fruits (on the paper, not real fruits) and hats (3 and 5). The activities were done in Berkeley and Stanford Elementary math circles (kids in grades 1 to 4). The handouts that were attached to Dave’s email are the ones used in the classes. The hats were made out of white paper and the green/yellow sticker dots were used. The activity with the fruits took 20 – 30 minutes and the hats took 50 minutes, but may take longer (in our class we had 25 kids and all of them were eager to play).

We had quite a few parents in the room. Some were helping the kids and some were shouting the answers out of the excitement.

The majority of the kids coming to Berkeley and Stanford Elementary math circles are there because they are good and interested in math, but occasionally we do get kids who do not belong.

The fruits and 3 hats were pretty easy to 90% of the kids. 5 hats were difficult only when all 3 hats had the same color.


Let’s not have a lecture about experiential learning! It’s the wrong format.

2) Who here as actually worked with kids on these problems? I’ve worked down to age 10 or so (Fifth grade), but not younger.

3) Some structure: I’ve done the fruit, but only in writing, not actually using fruit. The kids I’ve worked with (10 yrs old and up; voluntary after-school math club) love it, even in the written form. You don’t need bags of fruit to make it fun. I don’t know about younger kids. Also, we can use any objects–I myself would avoid fruit. I’d be afraid it would rot.

I’ve also done the ‘hats’. But I’ve used playing cards. They hold it to their foreheads and have to guess the color. They love it. Cheaper and easier than hats, and a million variations. (Again, I haven’t used this with really young kids).
a) I tell them how many black cards. Then they have to raise their hands if they have a black card. This goes quickly.
b) Go around the circle, telling how many black cards each person sees. Then raise your hand if you have a black card.
c) There have to be other variants right here that I haven’t thought of.
d) Liars and truth tellers: I tap each person once and they must tell the truth. But I tap one of the kids twice, and he or she must lie. Everyone knows that there is one liar. Then we go around and each tells (or lies about) how many black cards he or she sees.
e) Two liars
f) Liars, but they don’t know how many.

I don’t know enough yet about how (e) and (f) go.

Finally, I’ve done a more ‘free form’ logic puzzle, but it’s hard to do. Everyone gets a card, and I give clues: I see two spades, or three picture cards, or one card less than 5, or three red cards. Whatever, until they figure out their own card. My job is hard–keeping track both fo what I’ve said(!) and of all the possible deductions.

The kids ask for any of these activities–it’s a favorite.

4) A little deeper: I’d like to have a discussion about the intellectual structure. What are they learning? One one level, it’s simply deduction: “If she says she sees three black cards, and I only see two, mine must be black.”

Also–and I’m not yet clear on this–about the nature of truth values. A liar always utters a false sentence, and the notion that a false sentence can be evidence is unusual for them.

But I’m sure there is more, and I don’t have the full learning trajectory in mind yet. I would like to know how this builds into an understanding of the logical functions (and, or, etc), and maybe truth tables and the propositional calculus. I know that Smullyan has worked some of this out, but not for younger kids in math circles.

Anyone have thoughts along these lines?


Rodi, I like the wizard idea!

Marta, The links are very useful, thank you.

Let us discuss all the ideas together and start compiling a list.

Please let me know if you have any suggestions on the text of the handouts or if you have questions about the material.

What is the best way of communication: wiki? Google plus? Email? Phone?

Google plus allows up to 5 (depending on the computer power 10) participants with video calls (similar to skype, and allows more than 2 participants free).


Here are some organizational questions that come to my mind:

Would it be possible to send a correct link to wiki? The one referenced in your email, does not work.
To determine the amount of supplies needed, I need to know:

– How many children total will we have?
(I would guess 30 kids per session.)

– How many sessions/groups simultaneously will we have?
(Only one session at a time. In Houston the main group for primary school students
had many adults — I think 5 — in one room, and they had about 5 puzzles related to
parity. Each adult would help different groups of kids work on the puzzles.)

– How many kids in each group?
I’ll send the registration data closer to the workshop.)

Out of my experience the activity with the fruits takes 20 – 30 minutes. The activity with the hats takes at least 50 minutes. To compile a list of other activities, we need to know:

– How long is each session?

one hour

– How many times the same session will be repeated during the day?

(your session will happen twice — once in the morning and once in the afternoon. da)

Am I correct that you will provide the hats, the bags and the fruits? We had simple paper hats, constructed of a big piece of paper, with the yellow and green sticker dots. If it is a hat, is it OK to put it on the heads of several kids in a row or should we have the reusable hats?

(If you ask me to bring supplies, I will strive to bring, acquire and/or ship
the things that you need. There are many possibilities here. We could use paper
lunch bags and pictures of fruit for the fruit problems. We could use playing cards held up to foreheads as hats, or postit notes stuck on backs, or…



It says the age range for the activity is K to 3. The ideas sent around are great, but I do think they need to be modified and simplified for this age range. I’ve tried these sorts of logic games with kids as young as 5th grade, and even then it isn’t always successful. Kids get excited about the game and lose track of what information they have and what information they need to solve the puzzle. And if even one kid loses track and gives the wrong answer, it can throw the whole thing off.


Let me try to answer Mark’s questions.


I set up a registration form and people have started registering for our festival. Past experience shows that most registrations come
in the 3 days before the festival.

If you are interested, you may view the registration data at:

We have more registrations at this point than ever before. I think you should plan on having many kids.

The kids you will be working with are aged 4- 8. Parents will certainly be there. Someone, probably a parent saw the attached poster and registered the children as attending.

What will physically happen is up to you. In Houston last year, there was a group of facilitators who helped the young children
with various activities. The following link leads to some video from that room. You can get a general idea from the video. In order to
understand the specific activities, you would need to log in and read the attached PDFs.



I am hoping to get a group of 3-4 kids together this Thursday to try a few of these activities with. I plan to try the 3 games in the Logic Lesson handout from Laura. Are there any other particular activities any of you think I should try? My instinct is that games like the gnomes (Tanton’s video) will be too hard for this age group. At the festival, if we do games involving parity with groups bigger than 5 or so, we might want to split the kids into groups by age or experience. Not all will know the difference between even and odd, so we could have some games going that just address this concept for novices, and others that have a prerequisite knowledge of even versus odd. This is, of course, assuming that we are in a big room all together with 25 or more kids. With 7 of us, we could each do a game with 3-4 kids, or we could each partner up and take more kids. Maybe we could do one game first with a few kids volunteering, one adult leading (Laura?), and the rest watching. Then kids will say “I want to play too!” and at that point, we split up. Then maybe come together at the end if we think it’s productive.


I think, that as Rodi suggested, if we have less than 20 kids and we begin with the hats activity we can all start as one group for 1 or 2 games, let the kids observe the activity and then split.

If we have more than 20 kids, it maybe best to split first into smaller groups and then follow the above scenario.

Attached are lesson notes from 2 classes at Stanford and Berkeley Elementary math circles. The clever prince was a 10 minute class discussion. We received many interesting suggestion and came realty close to solving the problem, unfortunately we did not have enough time to finish this problem in class. Logic part 2 was more appropriate for the kids in grades 2 and 3, and was difficult for younger kids.


I wrote a story that weaves together all 4 of the games that Laura sent out. I did it today with a group of kids and it was very successful.. No one wanted the circle to end. The story is attached. There are a few lines in the story early on to help kids focus attention and feel confident in the group.
Below are some notes about some relevant things happened in the circle. I think it’s going to be a lot of fun to do these activites in DC. See below, and let me know what questions you might have about how it went today.

In attendance were 6 participating kids: E – age 5 (but almost 6),G and J – both age 7, S and A – both age 8, and V – age 8 about to turn 9. We also had 2 helping kids (C – age 11, and R – age 12) and 2 parents.

We read the story and played the games. I didn’t have time to make hats, so we used posterboard headbands as hats and colored small post-its as stickers. After doing the circle, I got feedback from everyone, and even did informal approval/disapproval voting.

The story took 2 minutes to read. Afterwards all kids reported thoroughly enjoying it, and they helped me clear up any ambiguities that were in it originally. For instance, I hadn’t originally had an ending for after the games, but the kids kept saying “tell us what happened!”)

Picking Fruit took 6 minutes with this group. Most said it was too easy, but the younger kids enjoyed it and experienced success, and all agreed it was worth keeping in.

Wearing Hats took 25 minutes. All liked it, and were vehement that it should be kept in. For one round, the older kids R and C wanted to join in. Most of the kids wanted chances to explain their thinking to the group, and this took a lot of time. G did not get her many opinions heard due to time constraints, and found this a bit frustrating. (I did give her time at the end to explain all her thoughts to me alone). This activity would be better with 3 to 5 kids.

The Very Clever Prince took 14 minutes. There was true collaboration in reaching a solution, and they did eventually come up with the idea of eating the thing. There was a lot of satisfaction from problem solving here. They said afterwards that even though it took so long, it was worth it. All the kids, even the older ones, and the parents, participated.

The Tiger or the Princess took 36 minutes. The group was able to quickly solve Case 1. Some of the kids did not know how to read, so even though I had printed out and taped up the signs, I had to read and reread them aloud many many times. It would be better to put up the signs with pictures somehow. Also, a number of the kids didn’t really get the initial premise that each room contains a princess or a tiger, and that there could be tigers in both or princesses in both. By the end of the game it was obvious that we should have drawn this out on the board. (We ended up writing T or P in a box under the number 1 or 2.) Case 2 was solvable, but a few of the younger kids left the circle at that point. Case 3 was too hard for all but V and R. A and G tried to follow along but were just too confused, and the others tried to come back but were too frustrated. I don’t think we should use Case 3 at all if we use this game.

I have one question about Case 3 sign 1. After I finished the story about the kingdom and dismissed the kids, V and R debated for quite some time about how to negate that statement to test what happens when it’s false. V’s contention is that you replace “or” with “and” to negate it. This is how I had done it beforehand. Rachel made a case for switching the location of the words “tiger” and “princess” in the statement to negate it. I argued that this doesn’t negate it, since I could say “do you want a cookie or a cupcake,” and then switch the order of cookie/cupcake but retain the original meaning. She argued that the the modifiers “this” and “the other” imply non-interchangable locations. I told them I would ask you all and find out for sure. (Again, I’d love a basic primer on strategies and axioms for such questions. I never advanced much beyond the contrapositive ) As I’m writing this tonight, I’m thinking that you could also negate that statement by replacing “or” with neither. So, help with that, please, whether or not we use this activity.

A few other things came up in our post-session discussion. Kids wondered whether it would be scary for younger kids. Our expert, E, said probably not as long as we’re clear that the life-and-death stuff is pretend. Small post-its were not the ideal sticker because they stick to each other so well that they are hard to separate quickly. If we use something like that, it would be good to have one leader talking and another leader sticking the stickers on the hats. Paper headbands instead of hats worked fine in this group, but each child needed a different size, so it wouldn’t be practical in a huge group. Paper hats are probably better. Some parents don’t like kids sharing hats (lice worries) so I had each kid initial her own with markers. FYI, we had no boys.

Finally, I think a good point to split into small groups would be as soon as we get to the picking fruit game. Everyone wanted a turn!

Do you know of a good way to introduce these rules in a math circle? There must be some sort of inquiry-type activity where people can derive them for themselves. Years ago, I taught courses to prep people for the LSAT (one of my jobs is for The Princeton Review), and we had real-life scenarios from which people could figure out the rules for themselves. One example is “If it’s raining, I bring my umbrella – what do you know for sure from that statement?”
Thanks again!

Thanks for the solutions – I’ll look over these.
In the meantime, I did a bit of editing on the story. During the sample class, the girls kept asking me “where is the queen?”
So I changed the ending to make the eldest daughter return and become ruling queen when the king retired, and I also fixed a few things with the words to make it flow better. The revised version is attached. [Called Story for Logic Games]
I also wanted to mention one other thing that came up in the class. When we played “Wearing Hats,” the youngest child (5 going on 6) didn’t really know how to figure out what color her dot was, and randomly guessed. I could tell she was just guessing randomly, so I put a piece of paper on the ground in the center of the circle (we were sitting on the floor), and stuck on it the same number and color of dots that were on the hats. Then she could really see how many dots there were of each color, and didn’t need to have that info in her head. No one else needed that. So we should be prepared to do this, or put them on the board, if we have really young kids in our groups.

HR Logic Lesson.pdf
Logic Lesson Notes.pdf
Logic games story.pdf
Story for Logic Games.pdf
Logic 2 solutions