Soft questions for 8 - 12 year olds

I am concluding my second year in mathematics at the university of Milan. I also happen to be an educator for 8 - 12 year old children (as a Scout).

Recently I have tried to fill some dead time by asking my kids some soft math/logic questions and having them argue about them among themselves.

Little did I know that my tiny project would have so much success that they now demand more questions, especially at lunch, and I have exhausted my store of puzzles.

I also found that while I sometimes manage to come up with a new one, it is often too easy or too hard for their age.

Does anybody know of some resource covering this kind of... thing?

Straight-up question suggestions are also welcome.

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Here is a sample of the questions I gave them (brutally stripped of the playful, narrative parts):

A certain frog doubles in size every day. When thrown into a well, it fills the well in 10 days. How many days does it take to fill a well, if I throw two frogs in it?

How high up can you count using only the 10 fingers of your hands? (Really "how many numbers can you represent", made them discover the binary counting).

How many fingers are there in one hand? And in two? And in ten? (evil trick question: almost everybody answers "a hundred" to the last question).

• Ten scouts set up a campsite. After finishing, each scout shakes hands with every other scout. How many handshakes were there in total? – NiloCK Jun 2 '16 at 2:38

An excellent ressource is Vladimir Arnold's list, which is (at least in part) available in a recent EMS Newsletter: http://www.ems-ph.org/journals/newsletter/pdf/2015-12-98.pdf (page 14).

A classical variant of your frog question is there, as well as many others. Some are said by Arnold to be easier for kids than for academics, and he might be right!

• When I clicked on the link, I noticed that the problems started on page not 14. I wonder if it depends on your browser. In any case, I enjoyed them especially number 11 - the classic, what color was the bear? – Amy B Jun 2 '16 at 3:01
• That was a gold mine. Thanks! – Riccardo Orlando Jun 2 '16 at 8:35
• @AmyB BK meant p. 14 as numbered in the newsletter; this actually appears on p. 16 of the PDF because two pages are taken up by the cover and front matter. Relatedly: You can link to a particular page in a PDF by using a hashtag/octothorp[e]/pound sign (#) followed by the PDF page; e.g., link. – Benjamin Dickman Jun 3 '16 at 0:01

Here are some of my favorites:

1. A man is given 3 pills and told to take them every half hour starting at 9 AM. When will he finish? (He'll finish at 10 AM and not at 10:30)
2. What number comes after ninety-nine? four hundred ninety-nine? four thousand ninety-nine?? (The answers are 100, 500, and 4,100 not 5000 for the last one)
3. A man has 17 sheep. All but 8 die of sickness. How many are left? (8 and not 9)
4. I have 2 coins totaling 35 cents. One of them is not a quarter. What are the coins? (A dime and a quarter - one is not a quarter but the other is).
5. I have 9 coins. One is counterfeit and is lighter than the others. I have a balance scale. What is the least number of weighings that I can use to find the counterfeit coin? (It can be done in 2 weighings)
6. A pencil and an eraser cost $1.10. The pencil costs one dollar more than the eraser. How much does the eraser cost? (5 cents and not 10 cents) Have fun. I will try to find a reference for these and others like it for you. • I wish I had a Math teacher like you. :) – Gaurav Agarwal Jun 1 '16 at 18:03 • Thanks! The coins on a scale one is probably too hard, and my evil children often find clever ways to circumvent the rules, or ignore those they find unfair - I can see them arguing that you can just try all the coins two by two on the scale, and completely ignore the minimising part. All the other are pretty good though, thank you! – Riccardo Orlando Jun 1 '16 at 23:08 • You can just ask them how to do it in 2 weighings (instead of mentioning minimizing). They weigh two groups of 3. If they are the same the third one has the counterfeit coin. If they are different, the lighter one has the counterfeit coin. Then take any two from the group with the counterfeit coin. Weigh any 2 and if they are the same the remaining one in the group is counterfeit and if not the lighter one is counterfeit. – Amy B Jun 3 '16 at 5:53 I think that you might find some good puzzles (some of which are quite active) in some Math Circles resources. Now, some of these (most?) will be more time than you would have available. But if you are reasonably creative you should be able to turn a lot of them into something that fits the dead time available. Good luck! Whenever I want to fill time with kids, simple reverse-problems come to my mind. Imagine a number, say 5. Which computation might lead to "5"? One might start with 2+3. That's right, let's see if they know another way. 1+4. Right again. :-) Maybe they come up with 3+2 and you may discuss whether this is the same as 2+3. In just searching for new ways of computing the result 5, you may find patterns (just as 2+3=3+2). To most of the kids I played this game with, the game was rather adaptive towards their level. The kids managed to search for possible answer on exactly their highest level (since more is not possible for them and less is boring to them). What about the following: • 1+1+1+1+1 • 6-1 • x+(5-x) • 5/2 + 5/2 •$\sqrt{5}^2\$
• ...

You can easily go on finding a way expressing 5 as an expected value in stochastics or as an integral or whatever. If you have many kids, this might create an interesting pool of creative and individual statements which need to be evaluated.

Although I think there is no clear learning goal, this may deepen kids understanding of numbers or whatever is involved.

When you look in a mirror, you notice that you are reflected right to left, that is, the writing on your shirt is backwards in your reflection. But the mirror doesn't reverse you from top to bottom. How can this be? Why does the mirror seem to care about right-left but not up-down?