The problems are meant to be solved by sequential trials beginning in the ones place and noticing the ramifications of carry as the student follows a logical progression and tests the results of each step against the "rules" or requirements of the problem.
One "standard" rule that would be known to the students but is not obvious to a random adult is that within a problem, each letter represents a different digit. It is unfortunate that this is not explicitly stated, but the truth is that nearly any randomly selected homework problem from grade school will depend on a local understanding of what constraints may be assumed and what approaches are to be used.
For problem C we have ABC + ABC + ABC = BBB
The student is expected to notice that
all the digits of the answer must be the same, and that
the middle digit of abc must match all the digits of bbb.
Third grade thinking for Problem C proceeds like this:
Try C = 1
1 + 1 + 1 = 3, so b would equal 3. For the tens place, that
means 3 + 3 + 3 would also need to have 3 in the digits place, but 3 +
3 + 3 = 9 so C cannot be 1.
Try C = 2
2 + 2 + 2 = 6, but 6 + 6 + 6 is 18, which can't be right because the
digits place has to be 6, because all three digits of the answer have
to be the same.
Try C = 3
3 + 3 + 3 = 9, but 9 + 9 + 9 = 27 when the ones place would have to be 9 to fit,
so that's wrong too.
Try C = 4
4 + 4 + 4 = 12, so now B would have to be 2 and we need to carry when
we test the tens digit: 2 + 2 + 2 (+ 1) = 7, which doesn't match 2, so
Try C = 5
5 + 5 + 5 = 15, so B would be 5 and there's another carry: 5 + 5 + 5
(+ 1) = 16, which is wrong because we need the ones place to be 5.
Try C = 6
6 + 6 + 6 = 18, and 8 + 8 + 8 (+ 1) = 25, but 5 is not 8 so this is
not the answer.
Try C = 7
7 + 7 + 7 = 21, so now we have to carry 2, and if B = 1 then 1 + 1 + 1
(+ 2) = 5, but we needed it to be 1, so this is wrong.
Try C = 8
8 + 8 + 8 = 24, so carry 2 again and try B = 4. Then 4 + 4 + 4 (+ 2) =
14. Hey! It matched! So then for the hundreds place we have to carry 1, so we need A to be a number that adds up to 4, and luckily 1 + 1 +
1 (+ 1) = 4, so YAY! This is correct!
Just to be sure, also try C = 9.
9 + 9 + 9 = 27, so carry 2 and B would be 7. Then 7 + 7 + 7 (+2) = 23,
and this is wrong because 3 is not equal to 7.
At a higher grade level, students would be expected to notice that
3(abc) = bbb
(bbb) / 3 = abc
and the middle digit of abc must be the same as all the digits of bbb.
Then the sequential trials step through all the possible values of bbb, and the thinking goes like this:
111 / 3 = 37 => too few digits
222 / 3 = 74 => too few digits
333 / 3 = 111 => wrong because 1 != 3
444 / 3 = 148 => correct!
555 / 3 => 185 => wrong because 8 != 5
666 / 3 = 222 => wrong because 2 != 6
777 / 3 = 259 => wrong because 5 != 7
888 / 3 = 296 => wrong because 9 !=> 8
999 / 3 = 333 => wrong because 3 != 9
= = = = =
Similarly, for problem B we have KLM + KLM = NKL.
EDIT: Thanks to those who pointed out that I had made an incorrect snap assumption that the final sum would increase monotonically with M; that of course is not true. I have now expanded the series. A third grader obviously would not make such a mistake. :)
The student should notice that when adding M + M,
the ones place digit becomes L. Then that same L
gets plugged into the tens place for the next column of addition,
so the answer to L + L has a tens place digit that becomes K.
The student also must notice that the first two digits of KLM
must match the last two digits of NKL.
The student also must notice that although the ones place and the tens place
can have answers larger than 9 (resulting in a carry), the hundreds place
must sum to less than ten, or else N would represent TWO digits,
rather than one. Another way of saying this would be that NKL can be
up to 999 but no larger.
Here's how the thinking proceeds:
Try M = 1
1 + 1 = 2, so L would be 2. Then 2 + 2 = 4, so K would be 4. Then 4 +
4 = 8, so N is 8. 421 + 421 = 842. The first two digits of 421 DO
match the last two digits of 842, so this is correct.
Try M = 2
2 + 2 = 4, so L is 4. Then 4 + 4 = 8, so K is 8. Then 8 + 8 = 16, so
N would be 16. 842 + 842 = 1684. The first two digits of 842 do match the last two digits of 1684, but N can't be higher than 9 or else it would not be a digit, so this is wrong.
Try M = 3
3 + 3 = 6, so L would be 6. Then 6 + 6 = 12, so K would be 2.
Then 2 + 2 (+1) = 5, so N = 5. Now we have 263 + 263 = 526.
The first two digits of 263 match the last two digits of 526, so this is correct.
Try M = 4
4 + 4 = 8, so L = 8. Then 8 + 8 = 16, so K = 6. Then 6 + 6 (+ 1) = 13, but N can't be greater than 9, so this has to be wrong.
Try M = 5
5 + 5 = 10 so L = 0. Then 0 + 0 (+ 1) = 1, so K = 1. Then N = 1 + 1 = 2. We have 105 + 105 = 210, and the first two numbers of KLM do match the last two numbers of NKL, so this is correct.
Try M = 6
6 + 6 = 12, so L = 2. Then 2 + 2 (+ 1) = 5, so K = 5. Then 5 + 5 = 10, but of course N can't be bigger than 9 so this is wrong.
Try M = 7
7 + 7 = 14, so L = 4. Then 4 + 4 (+ 1) = 9, so K = 9. Now that I'm getting the hang of this I can see that if K is 5 or more then N will be greater than the maximum digit of 9, so I can say this is wrong even without calculating that N = 9 + 9 = 18.
Try M = 7
7 + 7 = 14, so L = 4. Then 4 + 4 (+ 1) = 9, so K = 9. That means N will be 18 -- too big -- so this is wrong (even though the first two digits of KLM do match the last two digits of NKL).
Try M = 8
8 + 8 = 16, so L = 6. Then 6 + 6 (+ 1) = 13, so K = 3. Then 3 + 3 (+ 1) = 7, so N is 7. We have 368 + 368 = 736. The first two digits of KLM do match the last two digits of NKL, so this is correct.
Try M = 9
9 + 9 = 18, so L = 8. Then 8 + 8 (+ 1) = 17, so K = 7, and 7 + 7 (+ 1) = 15, which is more than a single digit, so this is wrong.
Try M = 10. Oops, M can't be larger than 9 because it wouldn't be a digit, so we are done.
Isn't it interesting that even though some of the answers were too big for N to squeeze in with the digits, it was still possible to make this pattern work for every value of M from zero to nine?
= = =
Problem A is DEF + FEF = GHH
This one bothers adults because it does not yield a single unique solution, but that won't bother most third grade students. Full credit would be given for any correct answer. This example demonstrates that the student can notice the odd-even implications of carry and the fact that carry from adding in the hundreds place will give too many digits.
The student figures out that F + F gives an answer where
the ones place digit is H. The observation that H in the digits place
will always be an even number, while H in the tens place can only be
an even number if there is no carry from the digits place (thus forcing
F to be less than 5) may be made at the outset or it may be discovered
along the way.
"F must be small enough that when you add F + F there's no carry,
because when you add E + E in the tens place it will also be
an even number, and if you then had to add a carried one,
you'd get an odd number, and then that couldn't match
the H in the ones place, which is even."
If this observation is not made, the student typically begins at the bottom
and works her way upwards. if it *is* made at the outset, the student
typically begins at 4 and works her way downward.
Third grade thought would be something like this:
F has to be less than 5, because 5 + 5 = 10 and that gives a carry digit, and I already know that since the tens place is E + E, I can't also have a carry digit and end up with an even number.
Try F = 4
4 + 4 = 8, so H is 8.
Now E cannot also be 4, since each letter in
this problem represents a different digit.
That means E has to be 9 because 9 + 9 = 18.
Now I have to carry one over to the hundreds
place, and I have to put 4 in for F in the hundreds place.
That means before I even put D in I already have
4 (+1) = 5,
so I have room to let D be anything less than 5 (because
if D were 5 or larger, the hundreds place wouldn't be a single digit).
But D cannot be 4 because that's already taken for F,
so D can be 1, 2, or 3.
But if D were 3 then we'd have G = 3 + 4 (+1) = 8, and H is already 8, so D can be 1 or 2, making G = 6 or 7.
So the answers (FHEDG) for F = 4 are 48916 and 48927
= = =
Now try F = 3
3 + 3 = 6, so H would be 6 and E + E would have to be 16, so E must be 8.
8 + 8 is 16 so there's a carry to the hundreds place.
I already know F is 3, and 3 (+1) = 4, so D can be anything from 0 to 5
without causing an overflow.
But D cannot be 3 because it's already used for F, so D can be 1, 2, 4, 5.
But if D were 2, G would be 6, and 6 is already used by H. And if D were 4, G would be 8, which is already used by E.
So D can be 1 or 5, which makes G = 5 or 9.
So the answers (FHEDG) for F = 3 are 36815 and 36859
= = =
Try F = 2
2 + 2 = 4 so H = 4. E can't be 2 because F is 2, so E must be 7 because 7 + 7 > = 14. That means there's a carry to the hundreds place.
Putting in 2 for F, we have 2 (+1) = 3 before we add D, so D can be anything up to 6 without giving too many digits.
But D cannot be 2 or 4 because they're already used, so D can be 1, 3, or 5.
But if D = 1 then G = 4, which is taken, so D can be 3 or 5.
So the answers (FHEDG) for F = 2 are 24736 and 24758
= = =
Try F = 1
1 + 1 = 2, so H must be 2. That means E must be 6 because 6 + 6 = 12. [Hey, E was 9, 8, 7, 6 when F was 4, 3, 2, 1!]
Since we have a carry, when we put 1 in for F in the hundreds place we get 1 (+1) = 2, which means D could be any number up to 7 without causing an overflow.
But D cannot be 1 or 2 or 6, because those are already taken, so D can be 3, 4, 5, or 7.
But if D is 4 then G is 6, which is taken by E, so D can be 3, 5, or 7
So the answers (FHEDG) for F = 1 are 12635, 12657, 12679
= = =
Try F = 0
0 + 0 = 0, and that's no good because F and H can't be the same. In fact, if F = 0 then everything has to be zero, which shows that zero isn't at all like other numbers.
So yes, it does make sense, and it can be solved by a third-grader who has been taught this approach to add and carry, and yes there are many other ways to approach this kind of problem.
In this context it's all about add and carry, and about noticing the impact of carry on the next column.
I give the results of Problem A as FHEDG because that's the order in which the child solves the problem.
I am new at this site and do not have the rep to comment, so I'll point out here that Phil's interesting perl solution doesn't fully apply the "standard rule" that for a given problem, each letter represents a different digit. Phil has applied that rule for D, E, F, and H, but has relaxed it for G, which seems a bit arbitrary. If multiple letters can represent the same number, we could also have things like FHEDG = 36336, right?