FOUR + ONE = FIVE



Solve the following cryptarithm:

      F O U R
   +    O N E
   ----------
      F I V E    
Determine the digit represented by each letter to make the addition correct.

E =
F =
I =
N =
O =
R =
U =
V =




Here is the answer:

There are 1,200 solutions 
(see below for the complete list).

Here is an explanation on how 
to solve sryptarithms:

(1) The letters can represent only 
one distinct value between 0-9.

(2) It is generally agreed that 
numbers can't begin with 0, 
for example, 0158.

(3) In most cryptarithms, there 
is only one solution.

(4) 19 is the maximum value with 
a carryover for two one-digit 
numbers in the same column.

Carry over can only be 1. For 
example, if you have the same 
letter in a column, that letter 
could be a nine and there could 
be a one from the previous 
carryover (that is how you 
could get 19).

(5) In many cases the result of 
an addition problem is one digit 
longer (in digit-length) than the 
addends – the numbers added. 
If there are only two addends, 
this implies that the extra digit 
is the number 1.

(6) In two addend cryptarithms, 
there may be columns that have 
the same letter in both the 
addends and also the result. 
If such a column is the units 
column, that letter must be 0. 
Otherwise, it can either be 0 or 
9 (and then there is a carry).  
If you have the same letter in 
just one of the addends and the 
result, then the other addend 
must be zero.


Now for the cryptarithm, 
  FOUR + ONE = FIVE:
(1) When I wrote out the letters 
that were used, only eight 
distinct letters occur but 
there are ten possible digits.  
This suggests that there are 
probably multiple solutions, 
and I don't like those because 
you must do more guess work.  
On the other hand, it may be 
easier to find a solution.  

(2) Look at the first column: 
    R + E + E.
Go back to rule six above: 
R must be equal to zero.  
In algebra, we call this the 
Addition Property of zero or 
Additive Identity Property: 
if you add zero to any 
quantity, you get that quantity.
It also tells us that E could 
be any number from 1 to 9.
	
(3) Look at column four: You 
have F in the top addend and 
you have F in the result.  
That tells us that there is no 
carryover or else there would 
be a different letter in the 
result.
	
(4) So, look at the previous 
column:  O + O = I.  
Since there can be no carryover, 
that tells us that O must be 1,
 2, 3, or 4.  It can't be 0 
 because we determined that 
 R must be 0.
It can't be 5, 6, 7, 8, or 9 
because that would cause us 
to have a carryover of one.
This also tells us that I must 
be 2, 3, 4, 5, 6, 7, 8, or 9 
(double the possibilties for 
O but then remember you could 
have a carryover of one from 
the previous column.

(5) We haven't narrowed things 
down too much but that is because 
there are so many solutions that 
work.  So, using the framework
above, take your first guess and 
work until you find a 
contradiction.  If there is no 
contradiction, then you have 
found an answer.  
(At this point, there are 241,920 
different combinations). 

Let's start with E = 1 and see 
where that takes us.  It doesn't 
affect any of the other columns, 
so we can look at either column 
2 or 3.

     F O U R
   +   O N E
   ----------
     F I V E   

E = 1
F =
I =
N =
O =
R = 0
U =
V =

I decided to try column three and 
begin with O = 2 (remember, it 
must be either 1, 2, 3, or 4, but 
we already selected E to be 1).  
If this leads to a contradiction, 
I will come back and try O = 3, 
then 4...  It has to be one of 
those numbers. 
So this tells me that I must equal 
4 or 5.  So I will start with 4.
Here is what we have so far:

      F 2 U 0
   +    2 N 1
   ----------
      F 4 V 1  

E = 1
F =
I = 4
N =
O = 2
R = 0
U =
V =

I know that F can be any number 
so I will leave that until the 
end and just assign what is left 
to F (since there will be two 
numbers left over, that should 
give me three solutions).
So look at the second column: 
  U + N = V.
I just need to find three numbers 
which satisfy that equation: 
  3 + 6 = 9 works.
And since U and N don't occur 
anywhere else, and since addition 
is commutative, I know that there 
will be two different solutions 
that work for that column and 
therefore there will be at least 
2 x 3 = 6 different solution.
So, I chose to make U = 3, N = 6, 
and V = 9.

Here is what we have so far:

      F 2 3 0
   +    2 6 1
   ----------
      F 4 9 1  

E = 1
F =
I = 4
N = 6
O = 2
R = 0
U = 3
V = 9

Everything works so far, so you 
can choose any of the three 
numbers left for F:  5, 7, 8.
So, here are six solutions 
from what we just did:

      5 2 3 0
   +    2 6 1
   ----------
      5 4 9 1  

E = 1
F = 5
I = 4
N = 6
O = 2
R = 0
U = 3
V = 9





      7 2 3 0
   +    2 6 1
   ----------
      7 4 9 1  

E = 1
F = 7
I = 4
N = 6
O = 2
R = 0
U = 3
V = 9


      8 2 3 0
   +    2 6 1
   ----------
      8 4 9 1  

E = 1
F = 8
I = 4
N = 6
O = 2
R = 0
U = 3
V = 9

Now switching U and N in the 
three solutions above, we get:

      5 2 6 0
   +    2 3 1
   ----------
      5 4 9 1  

E = 1
F = 5
I = 4
N = 3
O = 2
R = 0
U = 6
V = 9





      7 2 6 0
   +    2 3 1
   ----------
      7 4 9 1  

E = 1
F = 7
I = 4
N = 3
O = 2
R = 0
U = 6
V = 9


      8 2 6 0
   +    2 3 1
   ----------
      8 4 9 1  

E = 1
F = 8
I = 4
N = 3
O = 2
R = 0
U = 6
V = 9


When I started this explanation, 
I didn't know if we would run 
into a contradiction or not, but 
it turns out that we found six 
distinct solutions.  

Now, you could go back to my 
first guess, E = 1, and replace 
it by each of the digits 2, 3, 4, 
5, 6, 7, 8, or 9.  
Then proceed like I did above 
and you will wind up getting all 
the possible solutions.

After doing this exercise, I 
wondered how many different 
solutions that there were, so I 
used a brute force computer 
program to find out.  I just used 
eight nested FOR loops with 
values 0 to 9 and found out that 
there are 1,200 solutions to 
this problem.  

I checked and the six solutions 
that we came up with above are 
on the list of 1200 solutions.  




1200 solutions:
***    1230+256==1486
***    1230+257==1487
***    1230+259==1489
***    1230+265==1495
***    1230+267==1497
***    1230+268==1498
***    1240+296==1536
***    1240+297==1537
***    1240+298==1538
***    1250+236==1486
***    1250+237==1487
***    1250+239==1489
***    1260+235==1495
***    1260+237==1497
***    1260+238==1498
***    1260+274==1534
***    1260+278==1538
***    1260+279==1539
***    1260+283==1543
***    1260+287==1547
***    1260+289==1549
***    1270+264==1534
***    1270+268==1538
***    1270+269==1539
***    1270+293==1563
***    1270+294==1564
***    1270+298==1568
***    1280+263==1543
***    1280+267==1547
***    1280+269==1549
***    1280+293==1573
***    1280+294==1574
***    1280+296==1576
***    1290+246==1536
***    1290+247==1537
***    1290+248==1538
***    1290+273==1563
***    1290+274==1564
***    1290+278==1568
***    1290+283==1573
***    1290+284==1574
***    1290+286==1576
***    1320+354==1674
***    1320+358==1678
***    1320+359==1679
***    1320+374==1694
***    1320+375==1695
***    1320+378==1698
***    1340+352==1692
***    1340+357==1697
***    1340+358==1698
***    1340+385==1725
***    1340+386==1726
***    1340+389==1729
***    1350+324==1674
***    1350+328==1678
***    1350+329==1679
***    1350+342==1692
***    1350+347==1697
***    1350+348==1698
***    1350+392==1742
***    1350+396==1746
***    1350+398==1748
***    1360+382==1742
***    1360+385==1745
***    1360+389==1749
***    1360+392==1752
***    1360+394==1754
***    1360+398==1758
***    1370+324==1694
***    1370+325==1695
***    1370+328==1698
***    1380+345==1725
***    1380+346==1726
***    1380+349==1729
***    1380+362==1742
***    1380+365==1745
***    1380+369==1749
***    1390+352==1742
***    1390+356==1746
***    1390+358==1748
***    1390+362==1752
***    1390+364==1754
***    1390+368==1758
***    1420+436==1856
***    1420+437==1857
***    1420+439==1859
***    1420+453==1873
***    1420+456==1876
***    1420+459==1879
***    1420+473==1893
***    1420+475==1895
***    1420+476==1896
***    1430+426==1856
***    1430+427==1857
***    1430+429==1859
***    1430+462==1892
***    1430+465==1895
***    1430+467==1897
***    1450+423==1873
***    1450+426==1876
***    1450+429==1879
***    1450+473==1923
***    1450+476==1926
***    1450+478==1928
***    1450+482==1932
***    1450+486==1936
***    1450+487==1937
***    1460+432==1892
***    1460+435==1895
***    1460+437==1897
***    1460+472==1932
***    1460+475==1935
***    1460+478==1938
***    1470+423==1893
***    1470+425==1895
***    1470+426==1896
***    1470+453==1923
***    1470+456==1926
***    1470+458==1928
***    1470+462==1932
***    1470+465==1935
***    1470+468==1938
***    1470+482==1952
***    1470+483==1953
***    1470+486==1956
***    1480+452==1932
***    1480+456==1936
***    1480+457==1937
***    1480+472==1952
***    1480+473==1953
***    1480+476==1956
***    2150+196==2346
***    2150+197==2347
***    2150+198==2348
***    2160+185==2345
***    2160+187==2347
***    2160+189==2349
***    2160+194==2354
***    2160+197==2357
***    2160+198==2358
***    2170+184==2354
***    2170+186==2356
***    2170+189==2359
***    2170+194==2364
***    2170+195==2365
***    2170+198==2368
***    2180+165==2345
***    2180+167==2347
***    2180+169==2349
***    2180+174==2354
***    2180+176==2356
***    2180+179==2359
***    2180+194==2374
***    2180+195==2375
***    2180+196==2376
***    2190+156==2346
***    2190+157==2347
***    2190+158==2348
***    2190+164==2354
***    2190+167==2357
***    2190+168==2358
***    2190+174==2364
***    2190+175==2365
***    2190+178==2368
***    2190+184==2374
***    2190+185==2375
***    2190+186==2376
***    2310+347==2657
***    2310+348==2658
***    2310+349==2659
***    2310+374==2684
***    2310+375==2685
***    2310+379==2689
***    2310+384==2694
***    2310+385==2695
***    2310+387==2697
***    2340+317==2657
***    2340+318==2658
***    2340+319==2659
***    2340+351==2691
***    2340+357==2697
***    2340+358==2698
***    2350+341==2691
***    2350+347==2697
***    2350+348==2698
***    2350+364==2714
***    2350+368==2718
***    2350+369==2719
***    2350+391==2741
***    2350+396==2746
***    2350+398==2748
***    2360+354==2714
***    2360+358==2718
***    2360+359==2719
***    2360+381==2741
***    2360+385==2745
***    2360+389==2749
***    2360+391==2751
***    2360+394==2754
***    2360+398==2758
***    2370+314==2684
***    2370+315==2685
***    2370+319==2689
***    2380+314==2694
***    2380+315==2695
***    2380+317==2697
***    2380+361==2741
***    2380+365==2745
***    2380+369==2749
***    2390+351==2741
***    2390+356==2746
***    2390+358==2748
***    2390+361==2751
***    2390+364==2754
***    2390+368==2758
***    2410+453==2863
***    2410+457==2867
***    2410+459==2869
***    2410+463==2873
***    2410+465==2875
***    2410+469==2879
***    2430+461==2891
***    2430+465==2895
***    2430+467==2897
***    2430+485==2915
***    2430+486==2916
***    2430+487==2917
***    2450+413==2863
***    2450+417==2867
***    2450+419==2869
***    2450+463==2913
***    2450+467==2917
***    2450+468==2918
***    2450+481==2931
***    2450+486==2936
***    2450+487==2937
***    2460+413==2873
***    2460+415==2875
***    2460+419==2879
***    2460+431==2891
***    2460+435==2895
***    2460+437==2897
***    2460+453==2913
***    2460+457==2917
***    2460+458==2918
***    2460+471==2931
***    2460+475==2935
***    2460+478==2938
***    2470+461==2931
***    2470+465==2935
***    2470+468==2938
***    2470+481==2951
***    2470+483==2953
***    2470+486==2956
***    2480+435==2915
***    2480+436==2916
***    2480+437==2917
***    2480+451==2931
***    2480+456==2936
***    2480+457==2937
***    2480+471==2951
***    2480+473==2953
***    2480+476==2956
***    3140+156==3296
***    3140+157==3297
***    3140+158==3298
***    3150+146==3296
***    3150+147==3297
***    3150+148==3298
***    3210+257==3467
***    3210+258==3468
***    3210+259==3469
***    3210+265==3475
***    3210+268==3478
***    3210+269==3479
***    3210+275==3485
***    3210+276==3486
***    3210+279==3489
***    3210+285==3495
***    3210+286==3496
***    3210+287==3497
***    3240+276==3516
***    3240+278==3518
***    3240+279==3519
***    3250+217==3467
***    3250+218==3468
***    3250+219==3469
***    3260+215==3475
***    3260+218==3478
***    3260+219==3479
***    3260+281==3541
***    3260+287==3547
***    3260+289==3549
***    3270+215==3485
***    3270+216==3486
***    3270+219==3489
***    3270+246==3516
***    3270+248==3518
***    3270+249==3519
***    3270+291==3561
***    3270+294==3564
***    3270+298==3568
***    3280+215==3495
***    3280+216==3496
***    3280+217==3497
***    3280+261==3541
***    3280+267==3547
***    3280+269==3549
***    3280+291==3571
***    3280+294==3574
***    3280+296==3576
***    3290+271==3561
***    3290+274==3564
***    3290+278==3568
***    3290+281==3571
***    3290+284==3574
***    3290+286==3576
***    3410+452==3862
***    3410+457==3867
***    3410+459==3869
***    3410+462==3872
***    3410+465==3875
***    3410+469==3879
***    3420+451==3871
***    3420+456==3876
***    3420+459==3879
***    3420+471==3891
***    3420+475==3895
***    3420+476==3896
***    3450+412==3862
***    3450+417==3867
***    3450+419==3869
***    3450+421==3871
***    3450+426==3876
***    3450+429==3879
***    3450+462==3912
***    3450+467==3917
***    3450+468==3918
***    3450+471==3921
***    3450+476==3926
***    3450+478==3928
***    3460+412==3872
***    3460+415==3875
***    3460+419==3879
***    3460+452==3912
***    3460+457==3917
***    3460+458==3918
***    3470+421==3891
***    3470+425==3895
***    3470+426==3896
***    3470+451==3921
***    3470+456==3926
***    3470+458==3928
***    3470+481==3951
***    3470+482==3952
***    3470+486==3956
***    3480+471==3951
***    3480+472==3952
***    3480+476==3956
***    4130+156==4286
***    4130+157==4287
***    4130+159==4289
***    4130+165==4295
***    4130+167==4297
***    4130+168==4298
***    4150+136==4286
***    4150+137==4287
***    4150+139==4289
***    4150+176==4326
***    4150+178==4328
***    4150+179==4329
***    4160+135==4295
***    4160+137==4297
***    4160+138==4298
***    4160+192==4352
***    4160+197==4357
***    4160+198==4358
***    4170+156==4326
***    4170+158==4328
***    4170+159==4329
***    4170+182==4352
***    4170+186==4356
***    4170+189==4359
***    4170+192==4362
***    4170+195==4365
***    4170+198==4368
***    4180+172==4352
***    4180+176==4356
***    4180+179==4359
***    4180+192==4372
***    4180+195==4375
***    4180+196==4376
***    4190+162==4352
***    4190+167==4357
***    4190+168==4358
***    4190+172==4362
***    4190+175==4365
***    4190+178==4368
***    4190+182==4372
***    4190+185==4375
***    4190+186==4376
***    4230+286==4516
***    4230+287==4517
***    4230+289==4519
***    4260+271==4531
***    4260+278==4538
***    4260+279==4539
***    4270+261==4531
***    4270+268==4538
***    4270+269==4539
***    4270+291==4561
***    4270+293==4563
***    4270+298==4568
***    4280+236==4516
***    4280+237==4517
***    4280+239==4519
***    4280+291==4571
***    4280+293==4573
***    4280+296==4576
***    4290+271==4561
***    4290+273==4563
***    4290+278==4568
***    4290+281==4571
***    4290+283==4573
***    4290+286==4576
***    4310+372==4682
***    4310+375==4685
***    4310+379==4689
***    4310+382==4692
***    4310+385==4695
***    4310+387==4697
***    4320+351==4671
***    4320+358==4678
***    4320+359==4679
***    4320+371==4691
***    4320+375==4695
***    4320+378==4698
***    4320+395==4715
***    4320+396==4716
***    4320+398==4718
***    4350+321==4671
***    4350+328==4678
***    4350+329==4679
***    4350+362==4712
***    4350+368==4718
***    4350+369==4719
***    4360+352==4712
***    4360+358==4718
***    4360+359==4719
***    4360+391==4751
***    4360+392==4752
***    4360+398==4758
***    4370+312==4682
***    4370+315==4685
***    4370+319==4689
***    4370+321==4691
***    4370+325==4695
***    4370+328==4698
***    4380+312==4692
***    4380+315==4695
***    4380+317==4697
***    4390+325==4715
***    4390+326==4716
***    4390+328==4718
***    4390+361==4751
***    4390+362==4752
***    4390+368==4758
***    5130+146==5276
***    5130+148==5278
***    5130+149==5279
***    5130+164==5294
***    5130+167==5297
***    5130+168==5298
***    5140+136==5276
***    5140+138==5278
***    5140+139==5279
***    5140+186==5326
***    5140+187==5327
***    5140+189==5329
***    5160+134==5294
***    5160+137==5297
***    5160+138==5298
***    5160+182==5342
***    5160+187==5347
***    5160+189==5349
***    5170+192==5362
***    5170+194==5364
***    5170+198==5368
***    5180+146==5326
***    5180+147==5327
***    5180+149==5329
***    5180+162==5342
***    5180+167==5347
***    5180+169==5349
***    5180+192==5372
***    5180+194==5374
***    5180+196==5376
***    5190+172==5362
***    5190+174==5364
***    5190+178==5368
***    5190+182==5372
***    5190+184==5374
***    5190+186==5376
***    5210+263==5473
***    5210+268==5478
***    5210+269==5479
***    5210+273==5483
***    5210+276==5486
***    5210+279==5489
***    5210+283==5493
***    5210+286==5496
***    5210+287==5497
***    5230+261==5491
***    5230+267==5497
***    5230+268==5498
***    5260+213==5473
***    5260+218==5478
***    5260+219==5479
***    5260+231==5491
***    5260+237==5497
***    5260+238==5498
***    5270+213==5483
***    5270+216==5486
***    5270+219==5489
***    5280+213==5493
***    5280+216==5496
***    5280+217==5497
***    5310+372==5682
***    5310+374==5684
***    5310+379==5689
***    5310+382==5692
***    5310+384==5694
***    5310+387==5697
***    5320+371==5691
***    5320+374==5694
***    5320+378==5698
***    5320+394==5714
***    5320+396==5716
***    5320+398==5718
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Send any comments or questions to: David Pleacher