You probably think toys today are really cool compared to the bummer ones your parents must have had. Well, even though your parents lived in caves and ate dirt for dinner, we all had some cool toys ourselves – like the Mickey Mouse coin sorter, shown here. This toy hails from decades ago, or at least that’s when I got mine (let’s not focus on which decade…), and is both piggy bank and marvel of machinery. When you stick a coin into the slot at the top right, if it’s a quarter it’s heavy enough to tip the first red seesaw and fall into Mickey’s right arm. If it’s a nickel, it shoots straight through to his left arm. Pennies and dimes are narrower so they fall through a hole in that seesaw, then either tip or shoot through the bottom seesaw into the correct leg. So smart and simple, all without batteries, lasers or digital screens. Even without electricity or blinking lights, the money still adds up.

Wee ones: If you stick 2 quarters, 3 dimes, 2 nickels and a penny into Mickey, how many coins did you sort?

Little kids: If Mickey has 4 coins in each arm and 4 coins in each leg – 4 quarters, 4 dimes, 4 nickels, 4 pennies – how many coins is he holding?  Bonus: How many seesaw tips did those coins make happen? (Again, quarters and pennies each tip one seesaw; dimes and nickels shoot through.)

Big kids: If you put 5 of each type of coin into Mickey, how much money is that, in cents?  Bonus: If you swap out all the pennies and replace them with 5 extra quarters, now how much money do you have?

The sky’s the limit: If you put in 82 cents and exactly 4 seesaw tips happened, how many combination of coins could you have put in?

Answers:
Wee ones: 8 coins.

Little kids: 16 coins.  Bonus: 8 tips.

Big kids: 205 cents or $2.05, since it’s 5 times 41 cents.  Bonus: $3.25.

The sky’s the limit: 4 possibilities. You had to have put in 2 quarters (one tip each) plus 2 pennies for 2 more tips, so nickels or dimes make up the remaining 30 cents. Possible combinations are 3 dimes 0 nickels, 2 dimes 2 nickels, 1 dime 4 nickels, and 0 dimes 6 nickels, always with 2 quarters and 2 pennies.

Have you ever looked at a toy, book, or box of cereal and noticed a little rectangle of black stripes,  with little numbers under it? That’s called a bar code, and that symbol tells the scanner at the store which thing you’re buying, so the store can charge you. Every item has to have its own number so prices don’t get mixed up. And each digit has its own combo of 2 or 3 thin and thick bars, which the scanner can see and count up. Now there are QR codes which show a square full of teenier squares, and smartphones can snap a photo of them. But stripey ones really set the bar.

Wee ones:  If a bar code has 5 numbers in the first half and 5 in the second half, how many numbers does it have in total?

Little kids: If the digit 8 is shown using 2 fat bars and a 3 uses 1 fat bar and 1 thin, how many bars does the code 33333 88888 have?  Bonus: If a 4 needs 3 bars, how many bars for the code 43434 34343?

Big kids: If an 8 uses 2 fat bars and a 9 uses 1 fat and 2 thin, how many 8s are there in a code with 12 thin bars and 14 fat ones, if the code contains only 8s and 9s?  Bonus: What if the same bar code has 4 0s in the mix, where each 0 uses 2 thin bars – now how many 8s?

The sky’s the limit: If 10-digit codes can run from 00000 00000 to 99999 99999, how many codes have only odd digits for the first 5 digits?

Answers:
Wee ones: 10 digits total.

Little kids:  20 stripes, since it uses 10 digits each using 2 stripes.  Bonus: Now you have 25 stripes.

Big kids: There are enough thin bars for 6 9s, which use up 6 fat bars.  That leaves only 8 fat bars for 4 8s.  Bonus: The 0s use up 8 thin bars, leaving only 4 unclaimed thin ones, so now there are only 2 9s. That now leaves 12 fat bars for 6 8s.

The sky’s the limit: An odd first digit knocks out half the codes, and the next knocks out half of the remaining codes, and so on.  So only 1/32 of the 10,000,000,000 codes are possible.  That cuts you to 5 billion, then 2.5 billion, then 1.25 billion, then 625 million, then 312.5 million, or 312,500,000.  The other approach is that the first odd digit enables 5 families of codes, times 5 for each digit that follows…giving you 5x5x5x5x5x10x10x10x10x10.

It’s very cool that some letters and numbers look the same when we flip them upside-down. An 8 is a great example of that, as is a capital H. But other letters get a little more complicated, since there are two ways to flip them upside-down, and sometimes they will look the same only if you flip the correct way. Some letters and numbers have mirror symmetry: if you stand them up on a mirror, they basically look the same upside-down on the mirror as they do standing up. A capital B or E works that way. But the letter N doesn’t – it will look backwards. That’s because letters like N have rotational symmetry: if you spin them around so they’re upside down, they look the same. That won’t work on a B or E, as this time they’ll look backwards. The letter H will look right either way since it has both kinds of symmetry.  Check out the alphabet, and see how many words you can write that work either way.

Wee ones: Can the letter S look the same upside down? Which way do you have to flip it?

Little kids: If you write your name in all capitals and stand it on a mirror, how many of the letters will still look correct?  Bonus: How about if you spin your name halfway around on a piece of paper? Now how many letters still look right?

Big kids: Which uppercase letters of the alphabet look the same either flipped or spun upside down?  Bonus: How many letters in the alphabet look the same standing on their mirror image?

Answers:
Wee ones: Yes, if you spin it around. Standing the S on a mirror won’t work!

Little kids: Different for everyone…see if you can figure it out without a mirror!  Bonus: Different for everyone again.

Big kids: Not many…by our count there are only 4 of them: H, I, O, and X.  Bonus: All of those, plus B, C, D, E, and K (if written a certain way).