I have found in one of the Russian highschool competitions a relatively easy puzzle & to make it more challenging replaced a certain number
by N.
The modified text runs as follows:
Prove that among N sequential positive integers there always exists a number whose s.o.d. is a multiple of 11.
a. Restore N.
b. Now prove the statement.
From a computer run, N appears to be 29.
{Edit:
nope, this is incorrect, see Charlie's comment}
For a list of integers with s.o.d. being a multiple of 11, the differences between sequential numbers in that list is limited to the following values:
[3, 5, 7, 9, 12, 14, 16, 18, 21, 23, 25, 27, 29]
But I don't have a proof.
 Python code 
lastFound = 0
maxDiff = 0
diffList = []
for i in range(1,1000000):
if sod(i)%11 == 0:
diff = i  lastFound
if diff not in diffList:
diffList.append(diff)
lastFound = i
maxDiff = max(maxDiff,diff)
print(maxDiff)
print(sorted(diffList))
Edited on May 24, 2021, 12:27 pm

Posted by Larry
on 20210524 10:46:52 