13 - Large Sum

6 minute read Published: 2026-06-30

Problem statement on Project Euler's Website

Solution in the Uiua pad

Table of Contents

Problem

Work out the first ten digits of the sum of the following one-hundred $50$-digit numbers.

37107287533902102798797998220837590246510135740250 46376937677490009712648124896970078050417018260538 74324986199524741059474233309513058123726617309629 91942213363574161572522430563301811072406154908250 23067588207539346171171980310421047513778063246676 89261670696623633820136378418383684178734361726757 28112879812849979408065481931592621691275889832738 44274228917432520321923589422876796487670272189318 47451445736001306439091167216856844588711603153276 70386486105843025439939619828917593665686757934951 62176457141856560629502157223196586755079324193331 64906352462741904929101432445813822663347944758178 92575867718337217661963751590579239728245598838407 58203565325359399008402633568948830189458628227828 80181199384826282014278194139940567587151170094390 35398664372827112653829987240784473053190104293586 86515506006295864861532075273371959191420517255829 71693888707715466499115593487603532921714970056938 54370070576826684624621495650076471787294438377604 53282654108756828443191190634694037855217779295145 36123272525000296071075082563815656710885258350721 45876576172410976447339110607218265236877223636045 17423706905851860660448207621209813287860733969412 81142660418086830619328460811191061556940512689692 51934325451728388641918047049293215058642563049483 62467221648435076201727918039944693004732956340691 15732444386908125794514089057706229429197107928209 55037687525678773091862540744969844508330393682126 18336384825330154686196124348767681297534375946515 80386287592878490201521685554828717201219257766954 78182833757993103614740356856449095527097864797581 16726320100436897842553539920931837441497806860984 48403098129077791799088218795327364475675590848030 87086987551392711854517078544161852424320693150332 59959406895756536782107074926966537676326235447210 69793950679652694742597709739166693763042633987085 41052684708299085211399427365734116182760315001271 65378607361501080857009149939512557028198746004375 35829035317434717326932123578154982629742552737307 94953759765105305946966067683156574377167401875275 88902802571733229619176668713819931811048770190271 25267680276078003013678680992525463401061632866526 36270218540497705585629946580636237993140746255962 24074486908231174977792365466257246923322810917141 91430288197103288597806669760892938638285025333403 34413065578016127815921815005561868836468420090470 23053081172816430487623791969842487255036638784583 11487696932154902810424020138335124462181441773470 63783299490636259666498587618221225225512486764533 67720186971698544312419572409913959008952310058822 95548255300263520781532296796249481641953868218774 76085327132285723110424803456124867697064507995236 37774242535411291684276865538926205024910326572967 23701913275725675285653248258265463092207058596522 29798860272258331913126375147341994889534765745501 18495701454879288984856827726077713721403798879715 38298203783031473527721580348144513491373226651381 34829543829199918180278916522431027392251122869539 40957953066405232632538044100059654939159879593635 29746152185502371307642255121183693803580388584903 41698116222072977186158236678424689157993532961922 62467957194401269043877107275048102390895523597457 23189706772547915061505504953922979530901129967519 86188088225875314529584099251203829009407770775672 11306739708304724483816533873502340845647058077308 82959174767140363198008187129011875491310547126581 97623331044818386269515456334926366572897563400500 42846280183517070527831839425882145521227251250327 55121603546981200581762165212827652751691296897789 32238195734329339946437501907836945765883352399886 75506164965184775180738168837861091527357929701337 62177842752192623401942399639168044983993173312731 32924185707147349566916674687634660915035914677504 99518671430235219628894890102423325116913619626622 73267460800591547471830798392868535206946944540724 76841822524674417161514036427982273348055556214818 97142617910342598647204516893989422179826088076852 87783646182799346313767754307809363333018982642090 10848802521674670883215120185883543223812876952786 71329612474782464538636993009049310363619763878039 62184073572399794223406235393808339651327408011116 66627891981488087797941876876144230030984490851411 60661826293682836764744779239180335110989069790714 85786944089552990653640447425576083659976645795096 66024396409905389607120198219976047599490197230297 64913982680032973156037120041377903785566085089252 16730939319872750275468906903707539413042652315011 94809377245048795150954100921645863754710598436791 78639167021187492431995700641917969777599028300699 15368713711936614952811305876380278410754449733078 40789923115535562561142322423255033685442488917353 44889911501440648020369068063960672322193204149535 41503128880339536053299340368006977710650566631954 81234880673210146739058568557934581403627822703280 82616570773948327592232845941706525094512325230608 22918802058777319719839450180888072429661980811197 77158542502016545090413245809786882778948721859617 72107838435069186155435662884062257473692284509516 20849603980134001723930671666823555245252804609722 53503534226472524250874054075591789781264330331690

Foundation

IEEE 754 - Double precision

Uiua uses the double precision floating point standard of IEEE 754. This has the consequence of rounding integers once they get too large. At around $2^{53}$, or $9007199254740992$ things start breaking down.

Let's try all 1s but with one digit more than that. We'll enter that into the Uiua repl.

    11111111111111111 # our input
11111111111111112     # repl output

What about one of the 50 digit numbers?

    53503534226472524250874054075591789781264330331690
    53503534226472520000000000000000000000000000000000

That sure looks problematic.

Further reading

Double Precision Floating Point - Limitations on Int

Solution

Just trying it

An interesting observation to be made is that a 50-digit number seems to conserve the first 15-16 digits while rounding off the rest.

Since we want the first 10 digits, adding a mere 100 of them might not overflow the needed precision. Therefore we should just try adding them.

There is a lot of input. Therefore let's define the input only once, in the appendix. You can copy it from there if you wish. Just know that Input refers to that

Splitting a string by a character, and collecting the splits into an either boxes or just parsing numbers, is common. So common in fact that there are special aliases for this, ppbn and pbbn.

    ppbn@\n Input
    ⊜⋕⊸≠@\n Input
[
  37107287533902104000000000000000000000000000000000
  46376937677490010000000000000000000000000000000000
    ...and so on

Now we just have to reduce add and get the first 10 digits. We can unparse to turn the number into an array of digits, so that it works with take,10 or just under unparse take,10.

    ⍜°⋕ ↙₁₀ /+ ⊜⋕⊸≠@\n Input
5537376230

And that is the correct answer!

Not satisfying

This solution doesn't feel satisfying nor robust. Look at the mess beyond take,10.

    /+ ⊜⋕⊸≠@\n Input
5537376230390877000000000000000000000000000000000000

You can f.x. use Ruby, or some other programming language with BigInt, to parse each number and add them to get the following exact result.

# BigInt compared to Double Precision Float
5537376230390876637302048746832985971773659831892672
5537376230390877000000000000000000000000000000000000

Naive BigInt

Let's try to make our own BigInt. We know that we're only going to be doing addition so we can simplify by just focusing on that.

The problem we're having is the number type. What if we represent BigInt as an array of digits? Let's work out what we want.

Representation

It might make sense to have a function that transforms a string of digits into an array of digits.

    FromString ← ≡⋕

    FromString "18512831951230"
[1 8 5 1 2 8 3 1 9 5 1 2 3 0]

That also means we'd like to bring them back.

    ToString  ← /$"__"

    ToString [1 8 5 1 2 8 3 1 9 5 1 2 3 0]
"18512831951230"

Addition

We can't just mush together two arrays though. Plain arrays overflow.

    + [1 2 3 4] [3 4 5 6]
[4 6 8 10]

We need some magic method, BigAdd, that corrects the overflow.

    BigAdd [1 2 3 4] [3 4 5 6]
[4 6 9 0]

Well, we'd want to detect overflows, >9 and then do two things based on that.

  • Mark overflow with $1$, rotate and add to the next 10s place.
  • Mark overflow with $1$, subtract 10.

Here's a step by step. We'll start by identifying overflows.

    ⊸>9 [4 6 8 10]
[4 6 8 10]
[0 0 0 1]

We'll rotate the overflow by one so that the next digit is correctly incremented.

    ↻1 ⊸>9 [4 6 8 10]
[4 6 8 10]
[0 0 1 0]

# And if added
    + ↻1 ⊸>9 [4 6 8 10]
[4 6 9 10]

Now we just need the correction factor. Instead of correcting right away, we can fork the overflow array with the rotation and mul ¯10.

    ⊃(ׯ10|↻1) ⊸>9 [4 6 8 10]
[4 6 8 10]
[0 0 1 0]
[¯0 ¯0 ¯0 ¯10]

Adding these three together results in a correct BigInt addition.

    ++ ⊃(ׯ10|↻1) ⊸>9 [4 6 8 10]
[4 6 9 0]

Edge case - Adding various lengths

The [4 6 8 10] array was generated from two equal length arrays. What happens if we want to add together differing lengths? What about $1 + 999$?

    + [9 9 9] [1]
[10 10 10]

Hmm. That's strange, though intended. Recall the tutorial.

If one of the arrays has exactly one row, that array will be repeated for each row of the other array. This includes scalars.

Uiua.org | Tutorial on More Array Manipulation

This means that we should expect an error from $999 + 11$ unless the add function has a fill context.

    + [9 9 9] [1 1]
"Error: Shapes [3] and [2] are not compatible"

    ⬚0+ [9 9 9] [1 1]
[10 10 9]

Though... isn't that filling in from the left? We want the fill to come in from the right since [1 1] should affect the least significant digits first.

We could combat this by simply reversing the representations. Let's update our definitions.

    FromString ← ≡⋕⇌

    FromString "18512831951230"
[0 3 2 1 5 9 1 3 8 2 1 5 8 1]

Bringing it back is also a matter of reversal.

    ToString ← /$"__"⇌

    ToString [0 3 2 1 5 9 1 3 8 2 1 5 8 1]
"18512831951230"

Since we've reversed the notation, we also have to remember to rotate in the other direction as well.

Multiple steps needed

Let's take what we've gathered so far and start defining our BigAdd.

    BigAdd ← ++⊃(ׯ10|↻¯1) ⊸>9 ⬚0+

Let's add together $1234$ + $3456$ and see if the overflow is handled such that we get $4690.$

    BigAdd FromString "1234" FromString "3456"
[0 9 6 4]

    ToString BigAdd FromString "1234" FromString "3456"
"4690"

Alright. But what about something like $1 + 999$?

    ToString BigAdd FromString "1" FromString "999"
"9100"

We're clearly missing something. Let's make a function that checks if any digit is over $9.$ This can be a condition for a do loop that keeps correcting the digits until none are over the threshold. We'll update the BigAdd to contain such a loop.

    AddNotFin ← /↥ >9
    BigAdd    ← ⍢(++⊃(↻¯1|ׯ10)⊸>9|AddNotFin) ⬚0+

What does $999 + 1$ result in now?

    BigAdd FromString "999" FromString "1"
[1 0 0]

Huh. If we pass that to ToString we get back "001". It seems like the $10$s overflow correction was run multiple times but the last calculation rotated the $1$ such that it wrapped around. This makes sense though. The result should be 4 digits long but the longest array supplied is 3. We're simply missing a digit.

What if we just append a $0$, both(join 0), to the input arguments so that overflows have an empty significant digit to go to? If we're just adding together two numbers, at a time, the worst case is all $9$s, for both arguments, which still just requires one extra space.

We can then introduce a check on the other side that removes a zero if it is the most significant position. We can just check the last digit and switch between drop ¯1 or id.

Our definitions now look like this.

    AddNotFin ← /↥ >9
    RemZer ← ⨬(∘|↘¯1)=0⊸⊣
    BigAdd ← RemZer⍢(+ + ⊃(↻¯1|ׯ10)⊸>9|AddNotFin) ⬚0 + ∩(⊂0)

Let's test $1 + 999$, $999 + 1$ and $1234+3456$ again.

    ToString BigAdd FromString "1" FromString "999"
"001"

Oh, something went wrong. Let's work through our BigAdd a step at a time.

    ∩(⊂0) FromString "1" FromString "999"
[0 9 9 9]
[0 1]

Oh, we gotta keep our significant digits straight. Let's use backward join and try again.

    BigAdd ← RemZer⍢(+ + ⊃(↻¯1|ׯ10)⊸>9|AddNotFin) ⬚0 + ∩(˜⊂0)

    ToString BigAdd FromString "1" FromString "999"
"1000"
    ToString BigAdd FromString "999" FromString "1"
"1000"
    ToString BigAdd FromString "1234" FromString "3456"
"4690"

Great. That looks correct.

BigInt solution

Here are the definitions we've accrued so far.

    AddNotFin ← /↥ >9
    RemZer ← ⨬(∘|↘¯1)=0⊸⊣
    BigAdd ← RemZer⍢(+ + ⊃(↻¯1|ׯ10)⊸>9|AddNotFin) ⬚0 + ∩(˜⊂0)

For our input, we just want to end up with a row of strings. We change our partition function to use id instead of parse.

    ⊜∘ ⊸≠ @\n Input
╭─
  100×50 @
   digits
           ╯

# Let's look at the first one, just in case
    ⊢ ⊜∘ ⊸≠ @\n Input
"37107287533902102798797998220837590246510135740250"

Let's convert our array into BigInts.

    ≡FromString ⊜∘ ⊸≠ @\n Input
╭─
   100×50 ℝ
  0-9 μ4.532
             ╯

Then reduce them via BigAdd and converting back to string.

    ToString /BigAdd ≡FromString⊜∘ ⊸≠ @\n Input
"5537376230390876637302048746832985971773659831892672"

That's indeed the exact result. You can simply copy the first 10 digits or follow it up with take,10 and parse.

    ⋕ ↙₁₀ ToString /BigAdd ≡FromString⊜∘ ⊸≠ @\n Input
5537376230

Performance?

This does some shuffling so it might get slow with increasing input sizes. However, only a hundred 50-digit numbers is a breeze.

    ⊙◌⍜now(⋕ ↙₁₀ ToString /BigAdd ≡FromString⊜∘ ⊸≠ @\n Input)
0.00190925598144

Appendix

The input

In order to not explode the article, I will show this here once. You can copy this to use it.

Input = (
$ 37107287533902102798797998220837590246510135740250
$ 46376937677490009712648124896970078050417018260538
$ 74324986199524741059474233309513058123726617309629
$ 91942213363574161572522430563301811072406154908250
$ 23067588207539346171171980310421047513778063246676
$ 89261670696623633820136378418383684178734361726757
$ 28112879812849979408065481931592621691275889832738
$ 44274228917432520321923589422876796487670272189318
$ 47451445736001306439091167216856844588711603153276
$ 70386486105843025439939619828917593665686757934951
$ 62176457141856560629502157223196586755079324193331
$ 64906352462741904929101432445813822663347944758178
$ 92575867718337217661963751590579239728245598838407
$ 58203565325359399008402633568948830189458628227828
$ 80181199384826282014278194139940567587151170094390
$ 35398664372827112653829987240784473053190104293586
$ 86515506006295864861532075273371959191420517255829
$ 71693888707715466499115593487603532921714970056938
$ 54370070576826684624621495650076471787294438377604
$ 53282654108756828443191190634694037855217779295145
$ 36123272525000296071075082563815656710885258350721
$ 45876576172410976447339110607218265236877223636045
$ 17423706905851860660448207621209813287860733969412
$ 81142660418086830619328460811191061556940512689692
$ 51934325451728388641918047049293215058642563049483
$ 62467221648435076201727918039944693004732956340691
$ 15732444386908125794514089057706229429197107928209
$ 55037687525678773091862540744969844508330393682126
$ 18336384825330154686196124348767681297534375946515
$ 80386287592878490201521685554828717201219257766954
$ 78182833757993103614740356856449095527097864797581
$ 16726320100436897842553539920931837441497806860984
$ 48403098129077791799088218795327364475675590848030
$ 87086987551392711854517078544161852424320693150332
$ 59959406895756536782107074926966537676326235447210
$ 69793950679652694742597709739166693763042633987085
$ 41052684708299085211399427365734116182760315001271
$ 65378607361501080857009149939512557028198746004375
$ 35829035317434717326932123578154982629742552737307
$ 94953759765105305946966067683156574377167401875275
$ 88902802571733229619176668713819931811048770190271
$ 25267680276078003013678680992525463401061632866526
$ 36270218540497705585629946580636237993140746255962
$ 24074486908231174977792365466257246923322810917141
$ 91430288197103288597806669760892938638285025333403
$ 34413065578016127815921815005561868836468420090470
$ 23053081172816430487623791969842487255036638784583
$ 11487696932154902810424020138335124462181441773470
$ 63783299490636259666498587618221225225512486764533
$ 67720186971698544312419572409913959008952310058822
$ 95548255300263520781532296796249481641953868218774
$ 76085327132285723110424803456124867697064507995236
$ 37774242535411291684276865538926205024910326572967
$ 23701913275725675285653248258265463092207058596522
$ 29798860272258331913126375147341994889534765745501
$ 18495701454879288984856827726077713721403798879715
$ 38298203783031473527721580348144513491373226651381
$ 34829543829199918180278916522431027392251122869539
$ 40957953066405232632538044100059654939159879593635
$ 29746152185502371307642255121183693803580388584903
$ 41698116222072977186158236678424689157993532961922
$ 62467957194401269043877107275048102390895523597457
$ 23189706772547915061505504953922979530901129967519
$ 86188088225875314529584099251203829009407770775672
$ 11306739708304724483816533873502340845647058077308
$ 82959174767140363198008187129011875491310547126581
$ 97623331044818386269515456334926366572897563400500
$ 42846280183517070527831839425882145521227251250327
$ 55121603546981200581762165212827652751691296897789
$ 32238195734329339946437501907836945765883352399886
$ 75506164965184775180738168837861091527357929701337
$ 62177842752192623401942399639168044983993173312731
$ 32924185707147349566916674687634660915035914677504
$ 99518671430235219628894890102423325116913619626622
$ 73267460800591547471830798392868535206946944540724
$ 76841822524674417161514036427982273348055556214818
$ 97142617910342598647204516893989422179826088076852
$ 87783646182799346313767754307809363333018982642090
$ 10848802521674670883215120185883543223812876952786
$ 71329612474782464538636993009049310363619763878039
$ 62184073572399794223406235393808339651327408011116
$ 66627891981488087797941876876144230030984490851411
$ 60661826293682836764744779239180335110989069790714
$ 85786944089552990653640447425576083659976645795096
$ 66024396409905389607120198219976047599490197230297
$ 64913982680032973156037120041377903785566085089252
$ 16730939319872750275468906903707539413042652315011
$ 94809377245048795150954100921645863754710598436791
$ 78639167021187492431995700641917969777599028300699
$ 15368713711936614952811305876380278410754449733078
$ 40789923115535562561142322423255033685442488917353
$ 44889911501440648020369068063960672322193204149535
$ 41503128880339536053299340368006977710650566631954
$ 81234880673210146739058568557934581403627822703280
$ 82616570773948327592232845941706525094512325230608
$ 22918802058777319719839450180888072429661980811197
$ 77158542502016545090413245809786882778948721859617
$ 72107838435069186155435662884062257473692284509516
$ 20849603980134001723930671666823555245252804609722
$ 53503534226472524250874054075591789781264330331690
)