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## Tour the U.S.!

Published on Sunday, May 28, 2017 in , , ,

Starting in Delaware, you must tour the 48 contiguous United States, visiting each state exactly once. Where will you finish?

That's a puzzle from Futility Closet. I'll link directly to it later, so as not to get too far ahead. It's really a puzzle within a puzzle, however.

As anyone who has tried the Knight's Tour knows, moving around with the limitation of landing in each space only once can be quite a challenge.

The original Futility Closet puzzle can be solved either by thinking about that puzzle on its own with a little analysis. However, I think it might be a little more fun to try and solve the puzzle by trying various ways to get around the 48 contiguous United States, visiting each state exactly once.

To that end, I've written this as a puzzle you can play below. It's an interactive map, so you can scroll (by clicking and dragging outside of the U.S.) and zoom (using the + and - in the upper left corner of the map) as needed, which helps when you're trying access smaller states.

You start by clicking on any state. That state will turn blue, representing your current location. Some states will also appear in green, and these are states which border your current state, and to which you haven't already traveled.

To move to a new state, simply click on any one of the green states. Clicking on any other states won't have any effect. Once you click a green state, that state will become blue (again, denoting your current location), and a new set of bordering states will become green. Also, the state which you just left becomes red, which denotes that you can never move to that state again.

Quick side note: The way I've programmed it, you can't move directly from Arizona to Colorado or directly from New Mexico to Utah, as I didn't consider meeting at a single point to be a bordering state.

If you want to work on the original Futility Closet puzzle, zoom in on Delaware, and click on it to start. Otherwise, start on any state. Try and see if you can get to all 48 states just once each time. You'll get an alert if you've either solved the puzzle, or become trapped. In either case, the map will go blank and you can try again.

Even if you don't get all 48 states, try and beat your highest number of states each time. As you play, perhaps you'll even get a realization that will help you solve the original Futility Closet puzzle.

The more you try this, the better you'll get, and you'll probably be able to solve the 48-state U.S. tour puzzle and the Futility Closet puzzle before too long. At this point, you can challenge your friends, and show your skills at solving the tour. What kind of experience did you have solving the puzzle? I'd love to hear about it in the comments below!

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## The Collective Coin Coincidence

Published on Sunday, May 21, 2017 in , , , , , ,

This week, Diamond Jim Tyler demonstrates a new take on an old trick. Regular Grey Matters readers won't be surprised to learn that I like it because it's based on math, and it's very counterintuitive. We'll start with the new video, and then take a closer look at the trick.

This week's Scam School episode is called The Collective Coin Coincidence, and features Diamond Jim Tyler giving not only a good performance, but also a good lesson in improving a routine properly:

Brian mentions that this was an update from a previous Scam School episode. What he doesn't mention is that you have to travel all the way back to 2009 to find it! The original version was called The Coin Trick That Fooled Einstein, and Brian performed it for U.S. Ski Team Olympic gold medalist Jonny Moseley. It's worth taking a look to see how the new version compares with the original.

Brian and Jim kind of rush through the math shortly after the 4:00 mark, but let's take a close look at the math step-by-step:

Start - The other person has an unknown amount of coins. As with any unknown in algebra, we'll assign a variable to it. To represent coins, change or cents, we'll use: c

1 - When you're saying you have as many coins (or cents) as they do, you're saying you have: c

2 - When you're saying you have 3 more coins than they do, the algebraic way to say that is: c + 3

3 - When you're saying you have enough left over to make their number of coins (c) equal 36, that amount is represented by 36 - c, so the total becomes: c + 3 + 36 - c

Take a close look at that final formula. The first c and the last c cancel out, leaving us with 3 + 36 which is 39. If you go through these same steps with the amount of coins (in cents, as it will make everything easier) as opposed to the number of coins, it works out the same way. This is what Diamond Jim Tyler means when he explains that all he's saying is that he has \$4.25 (funnily enough, he says that just after the 4:25 mark).

As long as we're considering improvements, I have another unusual use for this routine. If you go back to my Scam School Meets Grey Matters...Still Yet Again! post, I feature the Purloined Objects/How to Catch a Thief! episode of Scam School, which I contributed to the show. It's not a bad routine as taught, but my post includes a tip which originated with magician Stewart James. This tip uses the Coin Coincidence/Trick That Fooled Einstein principle to take the Purloined Objects into the miracle class! I won't tip it here, so as not to ruin your joy of discovery.

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## Chinese Remainder Theorem II

Published on Sunday, May 14, 2017 in , , ,

Back in January of 2012, I wrote about the Chinese Remainder Theorem. Also, Martin Gardner taught the basics well in his book Aha!: Insight, including a trick where you can determine someone's secretly chosen number between 1 and 100 just from hearing the remainders when divided by 3, 5 and 7. Going over that post again, I've developed a few improvements to this trick that make it seem much more impressive, and maybe even easier to do.

The first problem is getting the remainders. With a standard calculator, it's not easy to do. The answer here is to simply have them divide the number by 3, 5 and 7, and have them tell you ONLY the number after the decimal point. Using the amount after the decimal point, you can work out the remainder. When dividing by 3, there are 3 possibilities for numbers after the decimal point:

• Nothing after the decimal point: Remainder = 0
• Number ends in .3333...: Remainder = 1
• Number ends in .6666...: Remainder = 2
To find the remainders when dividing by 5:
• Nothing after the decimal point: Remainder = 0
• Number ends in .2: Remainder = 1
• Number ends in .4: Remainder = 2
• Number ends in .6: Remainder = 3
• Number ends in .8: Remainder = 4
If you read my Mental Division: Decimal Accuracy tutorial, you'll get familiar with the 7s pattern. It's trickier than 3 or 5, but easily mastered. You only need to pay attention to the first 2 digits after the decimal point:
• Nothing after the decimal point: Remainder = 0
• Number ends in .14: Remainder = 1
• Number ends in .28: Remainder = 2
• Number ends in .42: Remainder = 3
• Number ends in .57: Remainder = 4
• Number ends in .71: Remainder = 5
• Number ends in .85: Remainder = 6
Using the decimals makes the trick seem more difficult from the audience's point of view, but it only requires a little practice to recognize which numbers represent which remainders.

The other improvement involves the process itself. Have them start by dividing their number by 7 and telling you the numbers after the decimal point. Using the steps above, you can quickly determine the remainder from 0 to 6. If the remainder is between 0 and 5, you can remember them by touching that many fingers of your left hand to your pant leg (or table, if present). For a remainder of 6, just touch 1 finger from your right hand to your pant leg or table.

What ever the remainder, imagine a sequence starting with this number, and adding 7 until you get to a number no larger than 34. For example, if the person told you their number ended in .42, you know the remainder is 3, and the sequence you'd think of is 3, 10, 17, 24 and 31 (we can't add anymore without exceeding 34). If the remainder was determined to be 4, instead, the sequence you'd think of would be 4, 11, 18, 25 and 32.

Next, ask the person to divide by 5, and tell you the part of the answer after the decimal point. Once you get this number, recall your earlier sequence (which can be recalled via the number of fingers resting on your pant leg or table), and subtract the new remainder from each number, until you find a multiple of 5. For example, let's say that when dividing by 7, their remainder was 4, so the sequence was 4, 11, 18, 25 and 32. Let's say that when dividing by 5, their remainder was 3. Which number from your initial sequence, when it has 3 subtracted from it, is a multiple of 5? is 4-3 a multiple of 5? No. Is 11-3 a multiple of 5? No. Is 18-3 a multiple of 5? YES! Now you have the number 18.

Whatever number you have at this point, is the smallest of 3 possible numbers. The other 2 possibilities are 35 more than that number and 70 more than that number. At this point, you can know that the chosen number is either 18, 53 (35+18) or 88 (70+18). The remainder when dividing by 3 will determine which one of these is the correct answer. For example, if they say their number, when divided by 3, ends in .3333..., you know the remainder is 1. So, run through all 3 numbers quickly and ask yourself which one is 1 greater than a multiple of 3. Is 18 - 1 a multiple of 3? No. Is 53 - 1 a multiple of 3? No. Is 88 - 1 a multiple of 3? YES! Therefore, their number must be 88.

There are certainly other approaches. In fact, a magician's magazine called Pallbearer's Review once presented this trick as a challenge, asking their readers to supply their methods. They received a wide variety of entries and many approaches. Most of them involved far more difficult methods than the above approach, which I prefer for actual performance.