# Upside-Down Number Problem

These are numbers that can also be read upside down as the same number, or different number: 0, 1, 6, 8, 9, 10, 11, 16, 18, so on. Each number can only be used once.

Using these numbers, try to get all numbers from 1-100.

EDIT: 1-100 has been completed by Marius, so keep going if you want to...

Here are 1 to 5 so you can understand.

1: 1

2: 8 - 6

3: 9 - 6

4: 10 - 6

5: 6 - 1

• did you delete the part saying we cannot use a number twice? if so we can do everything with 1+1+... Commented Oct 19, 2017 at 18:18
• do you mean each number, or each digit? e.g. is 11 valid? Commented Oct 19, 2017 at 18:21
• I'll allow that, also I added that to the original post.
– user41716
Commented Oct 19, 2017 at 18:23
• Why not 2 and 5? If we are talking traditional digital displays here, those two are definitely paired and ought to be included. Even handwritten, they can look quite similar when rotated. Commented Oct 19, 2017 at 23:07
• I was going to do that but there would be too many numbers, so I just decided to make it numbers that are upside-down in the font of the website. 1 isn't perfect but I allowed it anyways @feelinferrety
– user41716
Commented Oct 20, 2017 at 20:58

I love these puzzles.

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

• given other answers, I believe the question was not narrowly enough defined, but as I understand it, your answer is a possible one Commented Oct 20, 2017 at 8:52
• Good job, I edited the original post incase anyone wanted to keep going.
– user41716
Commented Oct 20, 2017 at 13:39
• $25 = 16+8−1$ ? Commented Sep 8, 2020 at 15:54
• But who would be so pedantic to check all of these anyway? Commented Sep 8, 2020 at 16:01
• @FlorianF. Thanks for the heads up. Obviously that should be +1 not -1. Do the rest check out? :) Commented Sep 9, 2020 at 5:42

Here are answers for all numbers in the range 0 to 249.

0 to 9 are:

0 = 0
1 = 1
2 = 8-6
3 = 9-6
4 = 9-6+1
5 = 6-1
6 = 6
7 = 6+1
8 = 8
9 = 9

For the rest, do this:

To these numbers you can add up to four multiples of ten expressed as (91-81), (96-86), (98-88) and (99-89), so it is then trivial to get any number in the range 0 to 49.
By adding multiples of fifty, expressed as (61-11), (66-16), (68-18), or (69-19) to these you get all numbers up to 249.

• Can we really flip '90' to '06'? (moot point, since you could still use e.g. 86-66 to get 20), and 61 would still be valid to open up everything above 60. Commented Oct 20, 2017 at 12:01
• That is a good point. The OP lists 10 as a reversible number, so I just assumed that 60, 80, 90 would be fine too. I'll edit my answer slightly. Commented Oct 20, 2017 at 13:16
• Any number that can be read upside-down can be used. Either the original one or the upside-down one
– user41716
Commented Oct 20, 2017 at 13:35

There only 5 numbers which can be used upside down. 0,1,6,8 and 9.

You can use any function

Ok then,

$$0=\log_ \frac 18 (\log_ 6 (((\sqrt9)!))$$

$$1=\log_ \frac 18 (\log_ 6 ((\sqrt{{\sqrt{\sqrt{(\sqrt9)!}}}})))$$

$$2=\log_ \frac 18 (\log_ 6 (\sqrt{\sqrt{\sqrt{\sqrt{\sqrt{{\sqrt{(\sqrt9)!)}}}}}}})$$

Keep adding sqrts over the sqrt(9)! to make all the other numbers. You also can add + 0 to use all the numbers.

• Good job, nice loophole. I checked it because it's correct, though it kind of defeats the purpose (I don't really care though)
– user41716
Commented Oct 19, 2017 at 19:02
• How do you get 1/2 from 0, 1, 8? Commented Oct 19, 2017 at 19:04
• Sorry, I will change the typo
– user35295
Commented Oct 19, 2017 at 19:16
• (You needn't accept the first "correct" answer given - you're welcome to accept whatever answer best exemplifies the solution you envisioned. And of course you're free to change your Accepted answer at any time. Not to take anything away from Allan's solution, but it seems that @Jaap Scherphuis' answer is more in line with what you were looking for, and there's no reason you couldn't accept that answer instead.)
– Rubio
Commented Oct 19, 2017 at 21:48

0, 1, 6, 8, 9 are given.

• 2 = 8 - 6
• 3 = 9 - 6
• 4 = 10 - 6
• 5 = 6 - 1
• 7 = 6 + 1
• 11 = 9 + 8 - 6
• 12 = 9 + 8 - 6 + 1
• 13 = 10 + 9 - 6
• 14 = 8 + 6
• 15 = 8 + 6 + 1
• 16
• 17 = 9 + 8
• 18
• 19
• 20 = 160 / 8
• 21 = 19 + 8 - 6
• 22 = 90 - 68
• 23 = 91 - 68
• 24 = 16 + 8
• 25 = 16 + 9
• 27 = 18 + 9
• 28 = 89 - 61
• 29 = 90 - 61
• 30 = 180 / 6
• 36 = 16 R * 9 (R is the square root key, common in most simple calculators)
• 37 = 98 - 61
• 38 = 98 - 60

• Oh, I seem to have misinterpreted the question. Commented Oct 19, 2017 at 18:08
• @Siddhartha No problem, I edited it as it might not have been clear.
– user41716
Commented Oct 19, 2017 at 18:11

• 20 = 80 - 60
• 22 = 88 - 66
• 23 = 89 - 66
• 24 = 96 / (9 - 6 + 1)
• 25 = 100 / (10 - 6)
• 26 = 86 - 60
• 27 = 88 - 61
• 28 = 88 - 60
• 29 = 89 - 60
• 30 = 100 / (11 - 6) + 9 + 1
• I think the idea is to generate all numbers from 1 to 100, not only the upside down ones, with expressions that only use upside down numbers. For example, 5 = 6 − 1. (But maybe I'm wrong.) Commented Oct 19, 2017 at 18:05
• I expected to use each digit only once. Is that right? Otherwise, there is a trivial solution: 1 + 1 + 1 + ... + 1. Commented Oct 19, 2017 at 18:18
• I've just asked OP. I don't know if we can use each digit once, or each number once? Commented Oct 19, 2017 at 18:19
• Sorry, when I edited it I forgot to add that back. Yes, each number can only be used once.
– user41716
Commented Oct 19, 2017 at 18:21
• Can we use calculator key shorthands? Keying = repeats the last operation. For instance, 3+4= gives 7, 3+4== gives 11, 3+4=== gives 15, and so on. If so, parenthesis are forbidden, since common calculators don't have them. Commented Oct 19, 2017 at 19:03

OP Edit: Understood the question wrong.

Numbers which are also numbers upside down: $0, 1, 6, 8, 9,$

Hence, there are $5 + 4*5 = 25$ such numbers, all of which can be generated using a simple program.

Here are the numbers: 1 6 8 9 10 11 16 18 19 60 61 66 68 69 80 81 86 88 89 90 91 96 98 99 100

• I may have explained it badly. I edited it, read it again.
– user41716
Commented Oct 19, 2017 at 18:11