The Book of Numbers

All the numbers, placed in a curious order

A curious order on the natural numbers

Consider the natural numbers up to ten, but in the following curious order:

8 5 4 9 1 7 6 10 3 2 0

What order is this? Is it random? No, I have placed these numbers in a very definite and logical order, using an underlying order idea with which I am sure that you, gentle reader, are familiar. Can you discover the underlying rule?

Surely one often gets insight to a problem by thinking about it from different perspectives, and so perhaps it may help for me to write out the numbers in a different manner, like this:

eight, five, four, nine, one, seven, six, ten, three, two, zero

OK, well, …how could that possibly help? Am I trolling? Perhaps not every new perspective is helpful.

Let me mention that the underlying order idea here applies just as much with larger numbers, and we can extend the list to the numbers up to one hundred, or one thousand, or one million. The new larger numbers will generally appear between and amongst the original numbers, however, rather than after them. Here are the numbers up to one hundred, for example, ordered according to the same underlying rule, with the original numbers up to ten appearing in the same order as they did before, highlighted in bold:

8 18 80 88 85 84 89 81 87 86 83 82 11 15 50 58 55 54 59 51 57 56 53 52 5 40 48 45 44 49 41 47 46 43 42 4 14 9 19 90 98 95 94 99 91 97 96 93 92 1 100 7 17 70 78 75 74 79 71 77 76 73 72 6 16 60 68 65 64 69 61 67 66 63 62 10 13 30 38 35 34 39 31 37 36 33 32 3 12 20 28 25 24 29 21 27 26 23 22 2 0

What is this crazy order? Can you observe carefully and discover it? Think about it before reading on…

Did you find the underlying rule? Let me provide a further hint by making once again the same trolling suggestion I had made before, namely, writing the numbers out in words. Does it help?

eight, eighteen, eighty, eighty-eight, eighty-five, eighty-four, eighty-nine, eighty-one, eighty-seven, eighty-six, eighty-three, eighty-two, eleven, fifteen, fifty, fifty-eight, fifty-five, fifty-four, fifty-nine, fifty-one, fifty-seven, fifty-six, fifty-three, fifty-two, five, forty, forty-eight, forty-five, forty-four, forty-nine, forty-one, forty-seven, forty-six, forty-three, forty-two, four, fourteen, nine, nineteen, ninety, ninety-eight, ninety-five, ninety-four, ninety-nine, ninety-one, ninety-seven, ninety-six, ninety-three, ninety-two, one, one hundred, seven, seventeen, seventy, seventy-eight, seventy-five, seventy-four, seventy-nine, seventy-one, seventy-seven, seventy-six, seventy-three, seventy-two, six, sixteen, sixty, sixty-eight, sixty-five, sixty-four, sixty-nine, sixty-one, sixty-seven, sixty-six, sixty-three, sixty-two, ten, thirteen, thirty, thirty-eight, thirty-five, thirty-four, thirty-nine, thirty-one, thirty-seven, thirty-six, thirty-three, thirty-two, three, twelve, twenty, twenty-eight, twenty-five, twenty-four, twenty-nine, twenty-one, twenty-seven, twenty-six, twenty-three, twenty-two, two, zero.

What is this order?

Interlude