# Brain Teaser: How well can you trust your intution?

Intuition can be a powerful guide for solving problems, but sometimes it can lead us astray. This problem is based on one from Project Euler

Let us say that a number has an ascending digit sequence if, writing the number from left to right, each digit is greater than or equal to the previous one. For example, 234 and 788 would have ascending digits. Define a descending digit sequence analogously. 855 and 210 would be descending sequences.

Now consider the numbers from 1 to 99. For one and two digit numbers, every one is an ascending or descending sequence. What does your intuition tell you about the comparative number of ascending and descending sequences? Are there the same number of each or are there more of one than the other? Was your intuition correct?

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My brain hurts just reading this question.

FishGutsDale (2038)

From intuition alone, I think there would be more descending, due to the simple fact that numbers can end in 0 but not start with 0.

I would think they are the same. 0–9 are single unique digits and would not represent a series. If you work forwards and backwards, 10 – 19 has 1 descending series, 9 ascending series. 90 – 99 is the opposite – 9 ascending series, 1 descending series. As you move towards a midpoint, it balances in mirror image. 10s = 90s, 20s =80s 30s =70s, 40s = 60s, 50 = 5 ascending, 5 descending.

PandoraBoxx (17996)

I would divide the total number of ascending digits by the number descending digits, then multiply the result times the square of the hypotenuse.

Dilettante (833)

@Dilettante I hope that wasn’t meant to make sense…..

@firemadeflesh As I hope the original question wasn’t meant to make sense.

Dilettante (833)

@FireMadeFlesh You see, I was an English major…I began to get a splitting headache when I was halfway through reading the question. “I was an English major” is also my standby excuse whenever I’m confronted with car problems, DoItYourself home repair, scientific matters, electronic device malfunctions, etc.

Dilettante (833)

10 – 19 have 2 descending and 9 ascending (under the stated rules, 11 is both). 90 through 99 have 10 descending and 1 ascending (again, 99 is both). Therefore, like @FireMadeFlesh I think the presence of the 0 means there are more descending.

Unless we start treating single digit numbers as being two-digit numbers with the first digit being 0, in which case my intuition says there are an equal number of each.

MrItty (17371)

@Dilettante Fair enough. I have these issues all the time with my brother, an English teacher, when I try to reason with him mathematically. He isn’t bad at science, but he has trouble with logic and calculations.

My instinct was to treat single numbers as being two digits (01, 02) as @MrItty suggested. Not sure what to do with the numbers 22, 33, 44, 55, 66, etc… Then, like others have said there is the whole zero thing. My intuition responds to these thoughts with “There is more descending numbers.” I have no idea if i have intuited the right answer.

liminal (7761)

The intuitions here are better than mine. There are more descending sequences. My intuition kept insisting that the number in each sequence was the same.

Consider first the single digit numbers. These are both ascending and descending so at this point the number of each type of sequence is the same.

When we more to two digit numbers, every ascending sequence can be matched with a descending sequence by reversing the digits, 37 would match with 73. The same does not hold for descending sequences because of the “zero thing.” 30 matches to 3, but we have already counted 3. There are therefore 9 more descending sequences corresponding to the 9 numbers ending with zero.

@MrItty , The numbers 22,33, etc. are both ascending and descending.

We can extend this reasoning to larger numbers. For example, the number of descending sequences from 1000 to 9999 equals the number of ascending sequences from 1 to 9999, because every descending 4 sequence of digits without zeros maps one-to-one with an ascending 4 digit sequence and every descending sequence ending in zeros matches one-to-one to an ascending sequence with a smaller number of digits. 3210 matches to 123 and 7400 matches to 47.

@Dilettante : You said hypotenuse!

TheLoneMonk (2892)

@LostInParadise er, yes, they are… I believe I said that. confused what you’re getting at by saying it again, prefaced by my name…

MrItty (17371)

@TheLoneMonk SEE…I TOLD you so! I was right all along.

Dilettante (833)

yes

Dr_Lawrence (19649)