Puzzle 36 - Solution workings

Key to colouring: Newly worked out - that line,
Previously worked out.
 DONALD
+GERALD
=ROBERT
 5ONAL5
+GERAL5
=ROBER0
     1 		Given that D = 5
Therefore T = 0 (5 + 5 = 10)
 5ONAL5
+G9RAL5
=ROB9R0
 11 11 		R must be an odd number because of the 1 carried over (L + L + 1 = R), So R = 1, 3, 7 or 9
Also E = 9 because it is the only number that will allow O to be part of the answer. I.e. any number plus 10 is for the only way it can appear in the answer but to allow this N+R must now equal B+10 so a one gets carried over.
 5ON4L5
+G9R4L5
=ROB9R0
 11 11 		If E = 9, A+A cannot equal an odd number (answer must be divisable by [2  A + A = 2xA]) so L+L+1 must equal R +10 (ie a ten has to carried from the tens into the hundreds to make the answer an odd number)
2A+1 can only be = 9 or 19
2A = 9 - 1 or 19 - 1
2A = 8 or 18
but E = 9 so A = 4 only
 5ON4L5
+1974L5
=7OB970
 11 11 		5+G+1= R must be less than 10 (nothing carried), therefore G = 0 - 4 (but T = 0 and A = 4) so 1, 2, or 3
when substituted in R = 7, 8 or 9, but R must be an odd number and E = 9 so R = 7 and therefore G = 1.
 5ON485
+197485
=7OB970
 11 11 		R=7 so 2L+1=7+10 so 2L=16 --> L=8
 5O6485
+197485
=7O3970
 11 11 		N+7 must be greater than 10 ie N = 3 or more [N >=3]
also N+7 = B+10 ie N = B+3, What's left? N cannot be = 4, 5, 7, 8, 9
If B=3 then N=6
if B=6 then N=9, but E=9 so B=3 only
 526485
+197485
=723970
 11 11 		1=G, 2=?, 3=N, 4=A, 5=D, 6=N, 7=R, 8=L, 9=E, 0=T
Therefore by exclusion O=2.
 526485
+197485
=723970
 11 11
Alternative

sent in by Alan Farrell, University of Texas at Brownsville

Prerequisite: High School Algebra

The solution is a bit lengthy and may be a little difficult to follow if you don't remember your algebra, but here goes.  First of all, we must assume that each letter represents a unique positive integer, in other words, two different letters may not represent the same integer. There are 10 letters and 10 integers (0-9).  Let each column of letters be denoted the numbers 1-6 from left to right. Since we cannot yet determine if a 1 must be carried over to the next column we add a row above DONALD labeled x1 x2 x3 x4 x5 x6. So we have

x1 x2 x3 x4 x5 x6
 D  O  N  A  L  D
+G  E  R  A  L  D
=R  O  B  E  R  T

where each x value is either 0 or 1.  Obviously x6=0 since column 6 is the first column to be added (if this is confusing you then try adding two 6 digit numbers and you'll see what I mean).

We are given that D=5, therefore we can immediately determine that T=0 and that x5=1.  So far we have

x1 x2 x3 x4  1  0
 5  O  N  A  L  5
+G  E  R  A  L  5
=R  O  B  E  R  0

Next we look at column 2 and note that the sum x2+O+E=O.  Since we do not know if x2 equals 0 or 1, we must consider both possibilities.  If x2=0 and O+E<10 then O+E=O and E=0, but T=0 so this cannot be true.  If O+E>10 then O+E=O+10 and E=10, but our variables must take values from 0-9 so this cannot be true.  Now we know that x2=1 since we eliminated the possibility x2=0.  With x2=1, if 1+O+E<10 then 1+O+E=O, or O+E=O-1 and E=-1, but this is negative so this cannot be true.  If 1+O+E>10 then 1+O+E=O+10, or O+E=O+9 and E=9.  So now we have:

x1  1 x3 x4  1  0
 5  O  N  A  L  5
+G  9  R  A  L  5
=R  O  B  9  R  0

From column 4 we have x4+A+A=9.  We cannot have x4=0 since then 2A=9 and A=4.5, which is not an integer.  Thus, x4=1 and 1+2A=9, which gives A=4, and x3=0.  So now we have

x1  1  0  1  1  0
 5  O  N  4  L  5
+G  9  R  4  L  5
=R  O  B  9  R  0

Next we look at column 1 and note that the sum x1+5+G=R.  It follows that x1+G=R-5 and thus 5<=R<=9 (since x1+G must be positive).  Now that we have a little information about R, we look at column 5 and note that the sum 1+L+L=1+2L=R+10 (because x4=1) .  This tells us that R is an odd number (2 times any number plus 1 is odd, try it if you don't believe me).  So R may only take the value 5, 7 or 9, but 5 and 9 are already taken, so R=7.  Since 1+2L>10 we have 1+2L=10+R, or 1+2L=17 and we find that L=8.So now we have

x1  1  0  1  1  0
 5  O  N  4  8  5
+G  9  7  4  8  5
=7  O  B  9  7  0

Since column 2 is greater than 10, x1=1.  Then from column 1 we get 1+5+G=7, thus G=1.  We are now left with the variables O, N, and B, and the integers 2, 3, and 6.  Column 3 gives N+7=B+10, or N-B=3.  Thus, N=6 and B=3.  Since 2 is the only integer left, we must have O=2.  Finally we have

 1  1  0  1  1  0
 5  2  6  4  8  5
+1  9  7  4  8  5
=7  2  3  9  7  0

We can now write this without the carried 1's and get

 5  2  6  4  8  5
+1  9  7  4  8  5
=7  2  3  9  7  0