Arrange the digits 0 to 9 such, that the number formed by the

first digit is divisible by 1, the number formed by the first two

digits is divisible by 2, that formed by the first three digits

divisible by 3, and so forth, thus the number formed by the first

9 digits will be divisible by 9 and that formed by all 10 digits

divisible by 10.

Here is what we would do.

_ _ _ _ _ _ _ _ _ _

those are my ten blanks.

Now let's start asking questions.

Since the whole thing has to be divisible by 10, the last number

must be a zero.

And since the first five digits must be divisible by 5, the fifth

digit must be a 5 or 0, but we have already used 0 so it must be a

five.

One last quickie is that the 2nd, 4th, 6th, and 8th and last

digits must be even, so all the rest are odd

So let's look at we have. I'll write the digits that could go in a

certain place under where they could go:

_ _ _ _ 5 _ _ _ _ 0

1 2 1 2 2 1 2 1

3 4 3 4 4 3 4 3

7 6 7 6 6 7 6 7

9 8 9 8 8 9 8 9

Now things get trickier.

Let's start with the 4th spot.

Any number is divisible by 4 if the last two digits are divisible

by 4

So what are the two-digit numbers divisible by 4 that begin with

an odd number that is not 5?

12 16 32 36 72 76 92 96

Hey, so the 4th digit is either a 2 or a 6.

Now, let's look at the 8th digit. If the first 8 digits are

divisible by 8, then they are divisible by 4 also. So if we apply

the same logic as above, we find that the 8th digit is either a 2 or

a 6.

So spaces 2 and 6 must be 4 or 8.

That narrows it down. So we have this:

_ _ _ _ 5 _ _ _ _ 0

1 4 1 2 4 1 2 1

3 8 3 6 8 3 6 3

7 7 7 7

9 9 9 9

Now for my next trick let's recall that every third space must be

divisible by 3. So every three numbers must sum to be divisible

by 3.

In particular spaces 4, 5, and 6 must sum to 3, and we already

have a five. So either we have

2 5 8 or 6 5 4

in positions 4, 5, and 6.

That is good stuff. That means we have either this:

_ 4 _ 2 5 8 _ 6 _ 0

1 1 1 1

3 3 3 3

7 7 7 7

9 9 9 9

or this:

_ 8 _ 6 5 4 _ 2 _ 0

1 1 1 1

3 3 3 3

7 7 7 7

9 9 9 9

Let's check each of the four digits we could put in the 8th place.

Since a number is divisible by 8 if its last 3 digits are, we need

number in the 6th, 7th, and 8th places to be divisible by 8. We see

that 816 and 896 work in the first case, and 432 and 472 work in the

second case. So we have:

_ 4 _ 2 5 8 _ 6 _ 0

1 1 1 1

3 3 9 3

7 7 7

9 9 9

or this:

_ 8 _ 6 5 4 _ 2 _ 0

1 1 3 1

3 3 7 3

7 7 7

9 9 9

Let's look at the 9th place now. Recall that 1+2+3+4+5+6+7+8+9 = 45,

so we don't need to worry about the 9th place - it will always work

out.

Let's look at the 1st and 3rd places. In case 1, only 147 and 741 are

options. In case 2, we could have 183, 189, 381, 387, 783, 789, 981, or

987. Note that 387 and 783 don't work, because then we don't have any

digits left for the 7th digit.

So our first case becomes:

_ 4 _ 2 5 8 9 6 3 0

1 1

7 7

We can just check the two numbers this can give rise to, namely 1472589630

and 7412589630. Since neither of them are divisible by 7, case one must be

a dead end. So we have to work with case two if we want a solution:

_ 8 _ 6 5 4 _ 2 _ 0

1 1 3 1

3 3 7 3

7 7 7

9 9 9

(and the first 3 digits are 183, 189, 381, 789, 981, or 987)

If we look at the 7th, 8th, and 9th digits to see if we can get them to sum

to three, we see that only 321, 327, 723, and 729 are options.

So which of our options for digits 1,2,3 and 7,8,9 can we use together? In

order to not repeat digits, we could use these:

183 and 729

189 and 327

189 and 723

381 and 729

789 and 321

981 and 327

981 and 723

987 and 321

This means that the only possible solutions to the puzzle are these:

1836547290

1896543270

1896547230

3816547290

7896543210

9816543270

9816547230

9876543210

This is now a small enough number of cases that we can check them by hand.

The first thing we should do is check the first 7 digits for divisibility by

seven. Here are the results:

1836547/7 = 262363.857142857...

1896543/7 = 270934.714285714...

1896547/7 = 270935.285714286...

3816547/7 = 545221

7896543/7 = 1128077.57142857...

9816543/7 = 1402363.28571429...

9816547/7 = 1402363.85714286...

9876543/7 = 1410934.71428571...

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