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# The root of the matter

## Question

This is a participatory puzzle. You must participate.

Write down three consecutive integers. There is no restriction, but your workload will be lighter if you pick small ones. Seriously. Write them down. I will wait.

Cube them - that will give you three larger, nonconsecutive integers. Now add their digits. Example: one of your original integers was 347. Remember I said it would be easier to use smaller ones? But does anyone listen any more? 3473 is 41,781,923. 4+1+7+8+1+9+2+3 = 35. Keep going. 3+5 = 8. OK now you can stop.

Do this for each of your three integers. Order the results, largest to smallest, to form a new three-digit number. Great job! You're all done. Except, now you have to read the spoiler.

Your new number is .... drum roll ... 981.

Your task is to debunk the rumor that I am psychic. Should not be hard.

## 3 answers to this question

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In any set of 3 consecutive numbers:

One of them will be a multiple of 3. When you cube the number, there will be a factor of 3*3*3. 3*3=9. All multiples of 9 have the property that the sum of their digits is divisible by 9.

One of them will be 1 less than a multiple of 3 (3x-1). All cubes of (3x-1) have the property that the sum of their digits is 8 (if you keep adding until you get a single digit). 1^3 =1. 9-1=8.
One of them will be 2 less than a multiple of 3 (3x-2). All cubes of (3x-2) have the property that the sum of their digits is 1 (if you keep adding until you get a single digit). 2^3=8. 9-8=1.

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This is a participatory puzzle. You must participate.

Write down three consecutive integers. There is no restriction, but your workload will be lighter if you pick small ones. Seriously. Write them down. I will wait.

Cube them - that will give you three larger, nonconsecutive integers. Now add their digits. Example: one of your original integers was 347. Remember I said it would be easier to use smaller ones? But does anyone listen any more? 3473 is 41,781,923. 4+1+7+8+1+9+2+3 = 35. Keep going. 3+5 = 8. OK now you can stop.

Do this for each of your three integers. Order the results, largest to smallest, to form a new three-digit number. Great job! You're all done. Except, now you have to read the spoiler.

Your new number is .... drum roll ... 981.

Your task is to debunk the rumor that I am psychic. Should not be hard.

if I choose -2, -1, 0, won't this fail? -2 cubed = -8, -1 cubed is -1, and 0 cubed is 0. Arranged from least to greatest is 0,-8,-1.

.

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This is a participatory puzzle. You must participate.

Write down three consecutive integers. There is no restriction, but your workload will be lighter if you pick small ones. Seriously. Write them down. I will wait.

Cube them - that will give you three larger, nonconsecutive integers. Now add their digits. Example: one of your original integers was 347. Remember I said it would be easier to use smaller ones? But does anyone listen any more? 3473 is 41,781,923. 4+1+7+8+1+9+2+3 = 35. Keep going. 3+5 = 8. OK now you can stop.

Do this for each of your three integers. Order the results, largest to smallest, to form a new three-digit number. Great job! You're all done. Except, now you have to read the spoiler.

Your new number is .... drum roll ... 981.

Your task is to debunk the rumor that I am psychic. Should not be hard.

if I choose -2, -1, 0, won't this fail? -2 cubed = -8, -1 cubed is -1, and 0 cubed is 0. Arranged from least to greatest is 0,-8,-1.
.

You're right.

The integers must be positive.

Good catch.

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