First, the user has given me six numbers on cards: 309, 41, 7, 68, 5, and 2. I need to form the smallest possible number using these cards. That means I have to arrange these numbers in such a way that when concatenated, they form the smallest numerical value.

The key here is to understand that I'm forming a single number by placing these cards side by side, so the order matters based on the numerical values. But since some are multi-digit numbers, I need to consider how to arrange them to minimize the overall number.

For example, if I just sort them numerically, I get: 2, 5, 7, 41, 68, 309. But if I concatenate that, it would be 25741309. But is that the smallest? Probably not, because the leading digit is 2, which is small, but I need to consider the place values.

This is similar to the problem of forming the smallest number from a set of numbers by arranging them in order. But I have to be careful with multi-digit numbers because their placement affects the digits.

A better approach might be to compare the numbers as strings when concatenated. For instance, I should compare different orders like 2,5,7,41,68,309 vs. 2,5,7,41,309,68, etc., but that seems inefficient with six numbers.

I recall that for such problems, a common strategy is to define a custom comparison. Specifically, for two numbers A and B, I should compare the concatenation AB vs BA, and choose the order that makes the smaller combined number.

For example, to decide whether 41 should come before or after 7, I compare 417 and 749? No, 41 and 7 concatenated would be 417 or 741. 417 vs 741 – 417 is smaller, so 41 should come before 7? But 417 is smaller than 741, so if I place 4 before 7, it makes 47 which is smaller than 74.

Let me formalize this.

Suppose I have two numbers X and Y. I need to decide whether to place X before Y or Y before X. I compare the numerical value of XY vs YX. If XY < YX, then X should come before Y; otherwise, Y before X.

For example, take X=41, Y=7. XY = 417, YX = 741. Since 417 < 741, so X should come before Y, meaning 41 before 7.

Similarly, for X=7, Y=5: XY=75, YX=57. 75 > 57, so YX is smaller, so Y should come before X, i.e., 5 before 7.

For X=68, Y=41: XY=6841, YX=4168. 6841 > 4168, so YX is smaller, so Y before X, i.e., 41 before 68.

For X=309, Y=2: XY=3092, YX=2309. 3092 > 2309, so YX is smaller, so Y before X, i.e., 2 before 309.

Now, I need to apply this pairwise comparison to sort all six numbers.

So, the numbers are: 2, 5, 7, 41, 68, 309.

I need to sort them using the comparison: for any two, compare XY and YX, and order such that XY < YX implies X before Y.

In the comparison, if XY < YX, then X before Y makes smaller concatenation.

But when sorting, I need a consistent order.

Let me list all pairs to see.

I can implement a sorting algorithm with this custom comparator.

Start with the list: [309, 41, 7, 68, 5, 2]

But I should have them as strings for easy concatenation.

Perhaps treat them as strings from the beginning.

So, the string representations: "309", "41", "7", "68", "5", "2"

Now, define a comparator: for two strings A and B, compare A+B vs B+A numerically, or lexicographically? Since it's numbers, better to compare numerically to be precise.

But in programming, often string comparison works if all are same length, but here different lengths, so numerical comparison is better.

For example, "2" and "10": if I compare "210" vs "102", numerically 210 > 102, so "102" is smaller, so "10" before "2"? "10" + "2" = "102", "2" + "10" = "210", 102 < 210, so "10" before "2" makes "102" which is smaller.

But numerically, 10 and 2, but when concatenated, the order matters.

In terms of the smallest number, "102" is smaller than "210", so yes.

Similarly, "5" and "2": "52" vs "25", 25 < 52, so "2" before "5".

Back to the list.

So, strings: "309", "41", "7", "68", "5", "2"

I need to sort this list using the comparator: A before B if (A+B) < (B+A) numerically.

Where < means numerical comparison.

Since all are strings, I can compare (A+B) and (B+A) as integers, but for large numbers, it might be big, but here it's manageable.

I can compare lexicographically if I pad with zeros, but it's easier to just compute the numbers.

Let me define the comparison function.

For two strings A and B:

Compute numA = int(A+B)

Compute numB = int(B+A)

If numA < numB, then A should come before B.

Else if numA > numB, B before A.

Else equal, but here all numbers different, so no equality.

Now, sort the list with this comparator.

I can do a bubble sort or something since only six elements.

List: ["309", "41", "7", "68", "5", "2"]

I need to arrange them so that when concatenated, the number is smallest.

Start comparing.

First, compare "2" and others.

"2" is small, but let's see.

I should find the smallest one to start, but with custom comparison.

Let me sort them.

Compare "2" and "5": "2"+"5" = "25", num=25; "5"+"2" = "52", num=52; 25 < 52, so "2" before "5".

Similarly, "2" and "7": "27" vs "72", 27<72, so "2" before "7".

"2" and "41": "241" vs "412", 241 < 412? 241 vs 412, 241 is smaller, so "2" before "41"? "2"+"41" = "241", "41"+"2" = "412", 241 < 412, yes, so "2" before "41".

But is that right? 241 is less than 412, so having "2" before "41" makes smaller number than "41" before "2".

Similarly, "2" and "68": "268" vs "682", 268 < 682, so "2" before "68".

"2" and "309": "2309" vs "3092", 2309 < 3092, so "2" before "309".

So "2" should come before all others? But that might not be optimal if there are smaller digits.

For example, "2" before "5", but "5" is larger, so "2" first seems good.

But let's check with "3" something, but "2" is the smallest digit, but since it's a single digit, it can be placed first.

In this case, since "2" compared to all others, "2" + other is smaller than other + "2", so "2" should be first.

Is that true?

"2" + "309" = "2309"

"309" + "2" = "3092" > 2309, yes.

Similarly for others.

But what about "5" and "2"? We have "2" before "5".

Now, similarly, "5" compared to others.

But since "2" is first, now arrange the rest.

List without "2": ["309", "41", "7", "68", "5"]

Now, compare "5" and "7": "5"+"7" = "57", "7"+"5" = "75", 57<75, so "5" before "7".

"5" and "41": "541" vs "415", 541 > 415? 415 < 541, so "41" + "5" = "415", "5" + "41" = "541", 415 < 541, so "41" before "5"?

Let's see: if "41" before "5", concatenation is "415", if "5" before "41", it's "541". Since 415 < 541, so "41" before "5" is better, meaning "41" should come before "5".

In the comparator: for A="41", B="5", A+B = "415", B+A = "541", since "415" < "541", so A before B, so "41" before "5".

Similarly, "5" and "68": "5"+"68" = "568", "68"+"5" = "685", 568 < 685? 568 vs 685, 568 < 685, so "5" before "68"? "5" + "68" = "568", "68" + "5" = "685", 568 < 685, so since A="5", B="68", A+B = "568" < B+A="685", so A before B, so "5" before "68".

But earlier with "41", we have "41" before "5".

Similarly, "5" and "7": "5"+"7" = "57", "7"+"5" = "75", 57<75, so "5" before "7".

"5" and "309": "5"+"309" = "5309", "309"+"5" = "3095", 5309 > 3095? 3095 < 5309, so "309" + "5" = "3095" < "5" + "309" = "5309", so "309" before "5"?

A="309", B="5", A+B = "3095", B+A = "5309", 3095 < 5309, so A before B, so "309" before "5".

But we also have "41" before "5", etc.

This is getting messy. Let me list all pairwise comparisons.

Make a table for each pair.

Denote the numbers: A="309", B="41", C="7", D="68", E="5", F="2"

But F="2" we already think should be first.

From earlier, F should be first.

Now, for the remaining: A, B, C, D, E

Compare E="5" with others.

E and C: "5"+"7" = "57", "7"+"5" = "75", 57<75, so E before C, i.e., "5" before "7"

E and B: "5"+"41" = "541", "41"+"5" = "415", 415 < 541, so "41" + "5" < "5" + "41", so B before E, "41" before "5"

E and D: "5"+"68" = "568", "68"+"5" = "685", 568 < 685, so "5" before "68"? 568 < 685, so E before D, "5" before "68"

E and A: "5"+"309" = "5309", "309"+"5" = "3095", 3095 < 5309, so A before E, "309" before "5"

Similarly, compare C="7" with others.

C and B: "7"+"41" = "741", "41"+"7" = "417", 417 < 741, so B before C, "41" before "7"

C and D: "7"+"68" = "768", "68"+"7" = "687", 687 < 768? 687 vs 768, 687 < 768, so "68" + "7" = "687" < "7" + "68" = "768", so D before C, "68" before "7"

C and A: "7"+"309" = "7309", "309"+"7" = "3097", 3097 < 7309, so A before C, "309" before "7"

Now, B="41" with D and A.

B and D: "41"+"68" = "4168", "68"+"41" = "6841", 4168 < 6841, so B before D, "41" before "68"

B and A: "41"+"309" = "41309", "309"+"41" = "30941", 30941 < 41309? 30941 vs 41309, 30941 < 41309, so "309" + "41" < "41" + "309", so A before B, "309" before "41"

A="309" and D: "309"+"68" = "30968", "68"+"309" = "68309", 30968 < 68309, so A before D, "309" before "68"

Similarly, A and C, we have A before C.

Now, summarize the comparisons:

• F="2" should be first, as it should come before all.

Now for the rest:

From above:

"309" (A) should come before "41" (B), because A before B

"309" before "7" (C)

"309" before "68" (D)

"309" before "5" (E)? From E and A, A before E, so "309" before "5"

But "5" is E.

Also, B before E, "41" before "5"

B before C, "41" before "7"

B before D, "41" before "68"

But A before B, so "309" before "41"

Now, what about "5" and "68": E before D, "5" before "68"

"5" before "7" from E before C

But "41" before "5", and "309" before "41", so "309" before "41" before "5"

Similarly, for "68" and "7": D before C, "68" before "7"

But "41" before "68", and "309" before "41", so "309" before "41" before "68"

Also "68" before "7", so "7" should be last or something.

Let's list the order.

From comparisons:

• "309" before "41" (A before B)

• "309" before "7" (A before C)

• "309" before "68" (A before D)

• "309" before "5" (A before E)

• "41" before "5" (B before E)

• "41" before "7" (B before C)

• "41" before "68" (B before D)

• "5" before "7" (E before C) ? E before C, "5" before "7"

• "5" before "68" (E before D)

• "68" before "7" (D before C)

Also, since "309" before all, and "41" before "5", "5" before "7", etc.

The order should start with "309", then "41", then "5", then "68", then "7"?

But we have "68" before "7", and "5" before "68", so after "41", "5" then "68" then "7".

But is "7" the last? We have six numbers, including "2" which is first.

F="2" is first.

Then A="309", then B="41", then E="5", then D="68", then C="7".

But let's confirm with the comparisons.

For example, between "5" and "68": we have "5" before "68"

Between "68" and "7": "68" before "7"

Also, "41" before "5", etc.

Now, what about "309" and "5": "309" before "5"

Similarly, all good.

But is there any number that should come before "309"? No, because "309" should come before others as per comparisons.

For example, "2" is before, then "309".

But let's check if "41" can be before "309"? No, from B and A, A before B, so "309" before "41".

Similarly, others.

So the order seems to be: "2", "309", "41", "5", "68", "7"

Now, concatenate: "2" + "309" + "41" + "5" + "68" + "7" = "2309415687"

But is this the smallest?

Perhaps I should consider other orders.

For example, what if I put "2" first, then "5", but from earlier, "5" should not come before "309" because "309" before "5".

Similarly, "41" before "5", but "309" before "41".

Another thing: "7" is last, but is there a better place?

What about "68" and "7": "68" before "7" is better.

Perhaps I can try the concatenation.

But let's think if "2" should be first. Since it's the smallest digit, and single digit, it makes sense to be first.

But in the number, the first digit is most significant, so having a small first digit is good, and "2" is the smallest possible first digit.

Similarly, after that, we need the smallest possible next digits.

In "2309415687", after "2", it's "309", which starts with 3, but is there a number that starts with smaller digit? "41" starts with 4, larger; "5" is 5, larger; "68" is 6, larger; "7" is 7, larger. So "309" has the smallest first digit after "2", starting with 3.

But "309" is 3, but there is "5" which is single digit, but "5" as a number is larger than "309" as a string? No, when placing, after "2", if I put "5", it would be "25...", but "25" is larger than "23" if I put "309" next? "2309..." vs "25...", but "25" is 25, while "2309" starts with 23, which is smaller than 25.

Numerically, 23 < 25, so "2309..." has a smaller prefix than "25...", so better to have "309" after "2" than "5".

Similarly, "41" starts with 4, 24... > 23..., so worse.

"68" 26... >23, "7" 27>23, so "309" is best after "2".

Then after "2" and "309", we have "309", so the number so far is 2309, now we need to add the next number.

The remaining are "41", "7", "68", "5".

We need to choose which one to add next to make the smallest continuation.

For example, if we add "41", it starts with 230941...

If we add "5", 23095...

If we add "7", 23097...

If we add "68", 230968...

Now, compare the first few digits: 230941, 23095, 23097, 230968.

230941 vs 23095: 230941 has more digits, but numerically, 230941 > 230950? 23095 is 23095, which is less than 230941? 23095 < 230941? No, 23095 is 23,095 and 230941 is 2,309,41? Better to think as numbers, but since it's a sequence, we need to see the string.

The current prefix is 2309, and we add a string, so the new prefix is 2309 + S for some string S.

S can be "41", "5", etc.

So the value is 2309 * 10^{len(S)} + int(S), but it's messy.

Compare the strings directly for the prefix.

For example, after "2309", adding "5" gives "23095"

Adding "41" gives "230941"

Now, "23095" vs "230941": compare character by character.

Position: 1:2=2, 2:3=3, 3:0=0, 4:9=9, then "23095" has fifth character 5, while "230941" has fifth character 4, and 4 < 5, so "230941" < "23095" because at fifth digit, 4 < 5.

"23095" is five characters, "230941" is six characters, so when comparing numbers, 23095 and 230941, 23095 < 230941? 23095 is about 23 thousand, 230941 is 230 thousand, so 23095 < 230941, but in the context, "23095" is the number 23,095, and "230941" is 230,941, which is larger, but for the overall number, we are building a multi-digit number.

I think I'm confusing myself.

The final number is a sequence of digits, so when we concatenate, "2" + "309" + "41" + ... = "230941..." which is a number with digits 2,3,0,9,4,1,5,6,8,7 so 10 digits: 2309415687.

If I put "5" after "2" and "309", it would be "23095" + "41" + "68" + "7" = "2309541687", which is 2,309,541,687.

Compare 2309415687 and 2309541687.

2309415687 vs 2309541687: both start with 2309, then 4 vs 5, 4<5, so 2309415687 < 2309541687, so better to have "41" before "5" in that position.

In our proposed order, after "309" we have "41", then "5", so "230941" then "5" makes "2309415", then "68" "68" makes "230941568", then "7" "7" makes "2309415687".

If after "309" I put "5" first, "23095", then "41" would be "2309541", which is larger than "230941" as above.

Similarly, if I put "68" after "309", "230968", then add others, "230968..." which starts with 230968, while 230941 is smaller.

"230968" vs "230941": at fourth digit 9=9, fifth digit 6 vs 4, 4<6, so "230941" < "230968".

Similarly, "23097" for "7" is larger.

So after "2" and "309", adding "41" is best because it starts with 4, which is smaller than 5,6,7.

"41" starts with 4, "5" with 5, etc.

But "41" is two-digit, but in terms of the digit sequence, after "2309", adding "41" gives digit 4, which is small.

Now, after "230941", we have to add the next number.

Remaining: "5", "68", "7"

We need to add one of them.

If add "5", get "2309415"

Then add "68", "230941568"

Then add "7", "2309415687"

If add "68" first, "23094168"

Then add "5", "230941685"

Then add "7", "2309416857"

Compare "2309415687" and "2309416857"

Both start with 230941, then 5 vs 6, 5<6, so "2309415687" < "2309416857"

Similarly, if add "7" first, "2309417", then add "5", "23094175", then "68", "2309417568", which starts with 2309417, while previous is 2309415, 5<7, so smaller.

So "2309415687" is better.

In this, we have "5" before "68" in the sequence.

But earlier in the pairwise, we had "5" before "68".

In the concatenation, "2309415" then "68" is "230941568", which has "5" before "68".

If I put "68" before "5", it would be "23094168" then "5" = "230941685", which is larger as above.

Similarly, "7" should be last.

But we have "7" and "68", with "68" before "7".

In the sequence, after "5" we have "68" then "7".

"2309415687" vs if "7" before "68", but "68" before "7" is better.

For example, "23094157" then "68" = "2309415768", compare to "2309415687": both 230941, then 5=5, then 6 vs 5? "23094156" vs "23094157", 6<7? Positions: "2309415687" has digits: pos1-10: 2,3,0,9,4,1,5,6,8,7

"2309415768" would be 2,3,0,9,4,1,5,7,6,8

Compare: up to 2309415 same, then 6 vs 7, 6<7, so "2309415687" < "2309415768", so better to have "68" before "7" as we have.

Similarly, all good.

But in the beginning, I have "2", then "309", but is "309" the best?

What about "5"? If I put "5" after "2", but "2"+"5" = "25", while "2"+"309" = "2309", and 25 < 2309? 25 is 25, 2309 is 2309, 25 < 2309, so "25..." is smaller than "2309...", so perhaps "2" then "5" is better?

I think I made a mistake here.

For example, if I start with "2" and then "5", concatenation "25", then add others.

But "25" is smaller than "2309" as a number, but since we are forming a multi-digit number, "25" followed by other digits might be smaller or larger depending on what follows.

For instance, suppose I have "2" + "5" + "309" + ... = "25309..." vs "230941..."

Now, "25309" vs "230941": both are numbers, 25,309 vs 230,941, and 25,309 < 230,941? 25 thousand vs 230 thousand, 25,309 < 230,941, yes, 25,309 is smaller.

25,309 vs 230,941: 25,309 < 230,941, so "25309..." < "230941...", so indeed, putting "5" after "2" is better than putting "309".

But earlier with custom comparison, I had "309" before "5", but that was for the order, but in that case, when "309" is before "5", it might not be optimal if "2" is first.

The custom comparison is for the relative order, but when "2" is fixed first, the comparison between "309" and "5" should be based on (309 + 5) vs (5 + 309) = 3095 vs 5309, and 3095 < 5309, so "309" before "5", meaning in the sequence, "309" then "5".

But if "2" is first, then "2" + "309" + "5" = "23095", while "2" + "5" + "309" = "25309", and 23095 vs 25309, 23095 < 25309? 23,095 vs 25,309, 23,095 < 25,309, so "23095" < "25309", so "2"+"309"+"5" is smaller than "2"+"5"+"309".

23095 and 25309, 23095 < 25309, yes.

But in terms of the full number, with other digits.

In "2"+"309"+"5" = "23095", then add "41" = "2309541", then "68" = "230954168", then "7" = "2309541687"

If "2"+"5"+"309" = "25309", then add "41" = "2530941", then "68" = "253094168", then "7" = "2530941687"

Now compare 2309541687 and 2530941687.

2309541687 vs 2530941687: both 10-digit numbers, compare digit by digit: 2=2, 3=3, 0<5, so 2309... < 2530..., so 2309541687 < 2530941687.

Thus, "2"+"309"+"5"..." is smaller than "2"+"5"+"309"..."

Similarly, if I put "2" and then "41", "241" then "309" = "241309", vs "2309309", but "241" > "2309" as numbers? 241 vs 2309, 241 < 2309? 241 is 241, 2309 is 2309, 241 < 2309, so "241..." < "2309..."? No, 241 is smaller than 2309, but "241309" vs "230941", for example.

"2"+"41" = "241", then "309" = "241309"

"2"+"309" = "2309", then "41" = "230941"

Now, 241309 vs 230941: 241,309 vs 230,941, 241309 > 230941, since 241 > 230, so "241309" > "230941", so better to have "2"+"309" before "41".

Similarly, all other numbers start with higher digits.

So "2" then "309" is best.

Then after that, as before.

But earlier with "5", we have "2"+"309"+"5" = "23095", which is 23,095.

Then we need to add the next.

The remaining are "41", "68", "7".

"41" starts with 4, "68" with 6, "7" with 7, so "41" is best to add next, giving "2309541"

Then remaining "68" and "7", "68" before "7" as per comparison, "686" vs "768" or in context, "230954168" then "7" vs "23095417" then "68", but "2309541687" vs "2309541768", compare 2309541687 and 2309541768, up to 2309541 same, then 6 vs 7, 6<7, so "1687" < "1768", so better to have "68" before "7".

Thus the order: "2", "309", "41", "5", "68", "7"

Concatenated: "2309415687"

But "5" is before "68", and "68" before "7".

In the list, we have "5" and "68", with "5" before "68".

But is there a better order?

What about putting "68" before "5"? But earlier comparison showed "5" before "68" is better.

For example, "5" and "68": "568" vs "685", 568 < 685, so "5" before "68".

Similarly, in the sequence.

Another thing: the number "7" is single digit, but since it's larger, it should be last.

Perhaps I should include it in the sort.

But let's confirm the full number.

So with order: F, A, B, E, D, C: "2", "309", "41", "5", "68", "7" -> "2309415687"

Numerical value: 2,309,415,687

But is this the smallest?

Another possibility: what if I put "2" first, then "5", but we saw that "2"+"5"+"309"..." is larger than "2"+"309"+"5"..."

Similarly, "2" then "41" is worse.

After "2" and "309", if I put "7", but "7" starts with 7, larger than 4 from "41", so worse.

"7" is "7", which as a string, when added, "23097" which is larger than "23094" from "41".

So no.

But we have "5" which is single digit, but after "2309", adding "5" gives "23095", adding "41" gives "230941", and since 23095 < 230941? 23095 and 230941, 23095 < 230941, so "23095" is smaller than "230941", but in the context, "23095" is followed by other digits, while "230941" is followed by other digits.

For example, "23095" + "41" = "2309541", but "2309541" vs "230941" + "5" = "2309415", and "2309415" vs "2309541": 2309415 and 2309541, both 7-digit, 2309415 < 2309541, since 415 < 541.

Similarly, "2309541" > "2309415".

In the sequence, if after "2309" I add "5", get "23095", then add "41", get "2309541", then add "68", "230954168", then "7", "2309541687"

If after "2309" I add "41", get "230941", then add "5", get "2309415", then add "68", "230941568", then "7", "2309415687"

Now compare 2309541687 and 2309415687.

2309541687 and 2309415687: both 10 digits, compare:

Pos1:2=2

2:3=3

3:0=0

4:9=9

5:5 vs 4, 4<5, so 230941... < 230954..., so 2309415687 < 2309541687.

Thus, adding "41" before "5" is better, as we have.

Similarly, for the last part.

Therefore, the smallest number is formed by the order: "2", "309", "41", "5", "68", "7" -> 2309415687

But the number is 2309415687.

But let's write it as a number: 2,309,415,687

But is this the smallest?

Another thing: in the cards, "309" has a zero, which might be good to have after high digits, but in this case, since it's at the beginning after "2", it starts with 3, which is fine.

Perhaps I can consider if "309" should be split or something, but no, the numbers are given as is, so I have to use them as whole cards.

For example, I can't split "309" into 3,0,9; I have to use it as "309".

Similarly for others.

So, I think this is the order.

But in the initial list, there is "7" which is single digit, and we have it last.

But let's see if putting "7" earlier could be better, but since it starts with 7, which is high, it should be later.

For example, after "230941", if I put "7" before "5", but "7" is "7", adding "7" gives "2309417", then adding "5" gives "23094175", then "68" "2309417568"

Compared to "2309415" then "68" "230941568", and 230941568 vs 2309417568, but different length? 230941568 is 9 digits so far? Let's not.

"2309417" is 7 digits, "23094175" is 8, etc.

Better to think the full 10-digit number.

With order "2","309","41","7","5","68" -> "230941756568" -> 2,309,417,565,68? Messy.

"230941756568" as string.

Compared to "2309415687" which is 10 digits: 2,3,0,9,4,1,5,6,8,7

"230941756568" would be 12 digits? No, each card: "2" is one digit, "309" three, "41" two, "7" one, "5" one, "68" two, total 1+3+2+1+1+2=10 digits: positions: "2","3","0","9","4","1","7","5","6","8" so 2309417568

"2309417568" vs "2309415687": both 10 digits, compare: 230941 same, then 7 vs 5, 5<7, so 2309415687 < 2309417568, so better to have "5" before "7" as we have.

Similarly, other orders.

Thus, the smallest is "2", "309", "41", "5", "68", "7" forming 2309415687.

But in the concatenation, it's "2309415687".

Now, to present the number.

But the question is to form the smallest number with the given cards, so I think 2309415687.

But let's confirm if there is a smaller one.

What about starting with "2" and then "5", but we saw it's larger.

Or "5" first? But "5" is larger than "2", so "2" should be first.

Similarly, "7" is larger.

So "2" must be first.

Then after "2", the smallest next digit sequence.

"309" starts with 3, "41" with 4, "5" with 5, "68" with 6, "7" with 7, so "309" is best.

Then after "2309", as before.

Another idea: what if I put "309" later to have smaller digits earlier, but since it has a zero, if I put it after high digits, it might be better, but in this case, because "2" is single digit, it allows "309" to be early.

For example, if I put "41" after "2", "241", then I need to place "309", but "241309" which is larger than "230941" as before.

Similarly, not better.

So I think 2309415687 is the smallest.

But in the order, we have "5" before "68", but "68" is 68, which might be placed before "5" if it makes smaller, but comparison shows not.

Perhaps I can list all possible permutations, but with 6! = 720, too many, not practical.

I think the custom sort is correct.

So the smallest number is 2309415687.

But let's write it properly.

The cards: 2, 309, 41, 5, 68, 7

Concatenated: 2 + 309 + 41 + 5 + 68 + 7 = 2309415687

As a number: 2,309,415,687

But for the answer, I think just the number is fine.

The problem says "form the smallest number", so 2309415687.

But is there a smaller one?

What about 2, 5, 7, 41, 68, 309? That would be 2574168309

Compare to 2309415687: 2574168309 vs 2309415687, 2.57e9 vs 2.30e9, 2309415687 < 2574168309.

Similarly, 2, 5, 41, 7, 68, 309: 2541768309 > 2309415687

2, 41, 5, 68, 7, 309: 2415687309 > 2309415687

2, 309, 5, 7, 41, 68: 23095741368? 1+3+1+1+2+2=10 digits: 23095741368? "2","309","5","7","41","68" -> "23095774168"? "5"+"7" = "57", no: concatenate: "2" + "309" = "2309" + "5" = "23095" + "7" = "230957" + "41" = "23095741" + "68" = "2309574168"

2309574168 vs 2309415687: 23095 vs 23094, 23094 < 23095, so 2309415687 smaller.

Similarly, other combinations seem larger.

Thus, 2309415687 is indeed small.

But in our number, it ends with 687, but since it's the least significant digits, it doesn't matter much.

So I think it's correct.