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We are looking for two natural numbers with two special properties:
Their total is 817.
If you cross out the last digit of one number, you get exactly the other number.
What are the two numbers?
Please scroll down for the solution!
Photo: Michael Niestedt / DER SPIEGEL
solution
The two numbers are
743
and
74
.
Obviously, one number must be three-digit and the other two-digit.
We write the numbers as
abc
and
ab
, where a, b and c are the digits of the numbers.
Because the sum of
abc
and
ab is
817, a must either be 7 (then there is a 1 noted from the addition of the two digits for the place in front of it) or a = 8.
No solution can exist for a = 6 or a <6, because the sum of
abc
and
ab
then cannot be greater than 800.
There can be no solution for a = 8 either, because
abc is
then greater than or equal to 800 and
from
greater
than or equal
to 80. Accordingly, the sum of both numbers would be at least 880 - but it must be 817.
So:
a = 7
.
We write:
7bc + 7b = 817.
We now look at which values are possible for the digits we are looking for, adding 7bc and 7b in writing to get 817.
So b + 7 must either be 11 if at the same time b + c <10.
Then: b = 4 and c = 3.
The two numbers we are
looking
for are then
743 and 74
.
Or b + 7 = 10, i.e. b = 3.
But then b + c has to result in 17, because the ones add up to 7 and a carryover of one is needed so that 3 + 7 becomes exactly 11.
For b + c = 17, however, there is no one-digit solution for c as long as b is 3.
Therefore there is only one solution for this task.
The idea for this puzzle comes from the book "Why cows like to graze in a semicircle" by Albrecht Beutelspacher and Marcus Wagner.
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