Problem 19 · 2009 Math Kangaroo
Stretch
Number Theory
digit-sumplace-value
A secret agent wants to crack a six-digit code. He knows that the sum of the digits in the even positions is equal to the sum of the digits in the odd positions. Which of the following numbers is the code? (Each ? stands for an unknown digit.)
Show answer
Answer: D — 12?9?8
Show hints
Hint 1 of 3
For each number, circle the digits in the 1st, 3rd and 5th spots, and box the digits in the 2nd, 4th and 6th spots.
Still stuck? Show hint 2 →
Hint 2 of 3
The known digits in one group might already be too big for the other group to ever catch up, even using a 9 in each blank.
Still stuck? Show hint 3 →
Hint 3 of 3
Look for the one number whose two groups CAN be made equal.
Show solution
Approach: compare the known digits in the two groups and see which one can balance
- Add up the known digits in the odd spots (1st, 3rd, 5th) and in the even spots (2nd, 4th, 6th), and remember a blank can be at most 9.
- In A, B, C and E one group's known digits are already so far ahead that even filling the other group's blanks with 9 cannot make them equal.
- In D the number is 12?9?8: the even spots give 2 + 9 + 8 = 19, and the odd spots are 1 + ? + ?, which reaches 19 when both blanks are 9 (1 + 9 + 9 = 19).
- Only option D can have its two groups equal, so the code is option D.
Mark:
· log in to save