Double The Other, Made up the Same

This is inspired by a problem I read today.

Find 2 9-digit numbers (in base-10 notation) such that one is double the other. Both of them are made up of the 9 non-zero digits once each. 123456789 can be one of them, but 122344566 can't be one of them.

I came up with an answer, but I am not sure if it is the only answer. I had a lot of fun finding it.

Tell me in the comments the 2 numbers you came up with, and what process you used. It doesn't have to be a rigorous proof. It could be a simple investigation based on a particular nice insight(that's what I did)

I'll share my answer once one of you comes up with it and says so in the comments.

Have fun!

(P.S.-For anyone who wants to check out the sources of my problems, Adventures In Problem Solving By Shailesh Shirali and Martin Gardner's 'The Colossal Book Of Mathematics' are awesome to start out with)

#NumberTheory

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

$abcdefghi = 9$ digit number $(1)$

$2(abcdefghi) = 9$ digit number $(2)$

$a \leq 4$

$1 * 2 = 2, 2 * 2 = 4, 3 * 2 = 6, 4 * 2 = 8, 5 * 2 = 0, 6 * 2 = 2, 7 * 2 = 4, 8 * 2 = 6, 9 * 2 = 8$ - last digits of $n * 2 = 2n$

Using these facts and the restriction on $a$ :

$123456789 * 2 = 246913578$ - has $1, 2, 3, 4, 5, 6, 7, 8, 9$ - answer $_1$

$132456789 * 2 = 264913578$ - has $1, 2, 3, 4, 5, 6, 7, 8, 9$ - answer $_2$

$123546789 * 2 = 247093578$ - has $2, 3, 4, 5, 7, 8, 9$ - not an answer

$123457689 * 2 = 246915378$ - has $1, 2, 3, 4, 5, 6, 7 ,8, 9$ - answer $_3$

$123456798 * 2 = 246913596$ - has $1, 2, 3, 4, 5, 6, 7, 8, 9$ - answer $_4$

$213456789 * 2 = 426913578$ - has $1, 2, 3, 4, 5, 6, 7, 8, 9$ - answer $_5$

$214356789 * 2 = 428713578$ - has $1, 2, 3, 4, 5, 6, 7, 8$ - not an answer

$213465789 * 2 = 426931578$ - has $1, 2, 3, 4, 5, 6, 7, 8, 9$ - answer $_6$

$213456879 * 2 = 426913758$ - has $1, 2, 3, 4, 5, 6, 7, 8, 9$ - answer $_7$

$213456798 * 2 = 426913596$ - has $1, 2, 3, 4, 5, 6, 9$ - not an answer

$312456789 * 2 = 624913578$ - has $1, 2, 3, 4, 5, 6, 7, 8, 9$ - answer $_8$

$314256789 * 2 = 628513578$ - has $1, 2, 3, 4, 5, 6, 7, 8$ - not an answer

$312546789 * 2 = 625093578$ - has $2, 3, 5, 6, 7, 8, 9$ - not an answer

$312465789 * 2 = 624931578$ - has $1, 2, 3, 4, 5, 6, 7, 8, 9$ - answer $_9$

$312457689 * 2 = 624915378$ - has $1, 2, 3, 4, 5, 6, 7, 8, 9$ - answer $_{10}$

$312456798 * 2 = 624913596$ - has $1, 2, 3, 4, 5, 6, 9$ - not an answer

$412356789 * 2 = 824713578$ - has $1, 2, 3, 4, 5, 7, 8$ - not an answer

$413256789 * 2 = 826513578$ - has $1,2, 3, 5, 6, 7, 8$ - not an answer

$412536789 * 2 = 825073578$ - has $2, 3, 5, 7, 8$ - not an answer

$412365789 * 2 = 824731578$ - has $1, 2, 3, 4, 5, 7, 8$ - not an answer

$412357689 * 2 = 824715378$ - has $1, 2, 3, 4, 5, 7, 8$ - not an answer

$412356879 * 2 = 824713758$ - has $1, 2, 3, 4, 5, 7, 8$ - not an answer

$412356798 * 2 = 824713596$ - has $1, 2, 3, 4, 5, 6, 7, 8, 9$ - answer $_{11}$

Therefore there are $\fbox{11}$ $9$ - digit numbers $abcdefghi$ that when doubled to make $2(abcdefghi)$ , both numbers have all the $9$ non-zero digits once each.

@Sachetan Debray

A Former Brilliant Member - 1 year ago

That's right. I didn't get that, though. This is what I got

123567849*2=247135698

The 'click' I used was that in the string of 9 non-zero numbers, you have exactly 5 odd numbers, and these must be present in the larger number too, so there must be at least 5 carryover 1s. i think you can generate many more numbers using that.

Sachetan Debray - 1 year ago

Can anyone write a code for this? It'll be fun to see.

Mahdi Raza - 1 year ago

@Pi Han Goh ?

Mahdi Raza - 1 year ago

from itertools import permutations
from math import factorial 

# Record the number of solutions found
total_number_of_solutions_found = 0

# The following line reads: Find all permutations of the digits 1 to 9, but the leading digit is at most 4 (otherwise, doubling it will be bigger than a 5-digit number)
for this_permutation in ( list(permutations([*range(1,10)], 9))[0: int(4/9 * factorial(9) )] ):

    # for this specific permutation of digits, we combine these digits into a 9-digit integer
    s = ''
    for item in this_permutation: s += str(item)

    # Don't forget to convert this string into an integer!
    concatenate_these_digits = int(s)

    # Now we double it!
    doubling_this_number = 2 * concatenate_these_digits

    # Once we double this number, we determine if this number has 9 distinct digits, and all its digits are non-zero
    if len(set(list(str(doubling_this_number)))) == 9 and (0 not in set(list(str(doubling_this_number)))):
        # If so, we have determined another solution
        total_number_of_solutions_found += 1

# Display the output
print(total_number_of_solutions_found) # Output: 18432

Pi Han Goh - 1 year ago

@Pi Han Goh – Clap Clap!! 👏👏

Mahdi Raza - 1 year ago

@Mahdi Raza – @Mahdi Raza, you mean 👏👏, right?

A Former Brilliant Member - 1 year ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$