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: 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
*** 5340+381==5721
*** 5340+386==5726
*** 5340+389==5729
*** 5360+381==5741
*** 5360+382==5742
*** 5360+389==5749
*** 5370+312==5682
*** 5370+314==5684
*** 5370+319==5689
*** 5370+321==5691
*** 5370+324==5694
*** 5370+328==5698
*** 5380+312==5692
*** 5380+314==5694
*** 5380+317==5697
*** 5380+341==5721
*** 5380+346==5726
*** 5380+349==5729
*** 5380+361==5741
*** 5380+362==5742
*** 5380+369==5749
*** 5390+324==5714
*** 5390+326==5716
*** 5390+328==5718
*** 5410+426==5836
*** 5410+427==5837
*** 5410+429==5839
*** 5410+462==5872
*** 5410+463==5873
*** 5410+469==5879
*** 5420+416==5836
*** 5420+417==5837
*** 5420+419==5839
*** 5420+471==5891
*** 5420+473==5893
*** 5420+476==5896
*** 5430+461==5891
*** 5430+462==5892
*** 5430+467==5897
*** 5430+482==5912
*** 5430+486==5916
*** 5430+487==5917
*** 5460+412==5872
*** 5460+413==5873
*** 5460+419==5879
*** 5460+431==5891
*** 5460+432==5892
*** 5460+437==5897
*** 5460+471==5931
*** 5460+472==5932
*** 5460+478==5938
*** 5470+421==5891
*** 5470+423==5893
*** 5470+426==5896
*** 5470+461==5931
*** 5470+462==5932
*** 5470+468==5938
*** 5480+432==5912
*** 5480+436==5916
*** 5480+437==5917
*** 6130+145==6275
*** 6130+148==6278
*** 6130+149==6279
*** 6130+154==6284
*** 6130+157==6287
*** 6130+159==6289
*** 6140+135==6275
*** 6140+138==6278
*** 6140+139==6279
*** 6140+153==6293
*** 6140+157==6297
*** 6140+158==6298
*** 6140+185==6325
*** 6140+187==6327
*** 6140+189==6329
*** 6150+134==6284
*** 6150+137==6287
*** 6150+139==6289
*** 6150+143==6293
*** 6150+147==6297
*** 6150+148==6298
*** 6150+174==6324
*** 6150+178==6328
*** 6150+179==6329
*** 6150+192==6342
*** 6150+197==6347
*** 6150+198==6348
*** 6170+154==6324
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