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
*** 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
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