Example Guide

Step 1: All three digits revealed Clue 4 tells us that all three digits in the test number (3, 0, and 2) are actually in our secret code. One of them is already in the correct position, while the other two appear in the code but need to be moved to different positions.

Step 2: Where can 3 be positioned? Looking at Clue 3, the test number 563 has one digit correct but in the wrong position. We know 3 is in our code, and here it's in position 3 marked as "wrong position." This means 3 cannot be in position 3 of our secret code. Therefore, 3 must be in either position 1 or position 2.

Step 3: Testing if 3 is in position 2 If 3 were in position 2, our code would be either 032 or 230. For Clue 4's test number 302, both 032 and 230 would result in all three digits being in wrong positions, not the required "1 correct and in the correct position, 2 correct but in the wrong positions." Therefore, 3 cannot be in position 2.

Step 4: Therefore 3 must be in position 1 With 3 in position 1, our code could be either 302 or 320. Checking against Clue 4: if the code were 302, all three digits would match their test positions exactly, giving us "3 correct and in the correct positions," which is not what the clue says. But if the code is 320, then 3 is in the correct position (position 1), while 0 and 2 are in wrong positions, giving us exactly "1 correct, 2 wrong" as required. Therefore, our code must be 320.

Step 1: Eliminate many digits at once Clue 4 tells us there are zero even digits in our code, which eliminates 0, 2, 4, 6, and 8. Clue 3 adds that 9, 4, and 2 are all incorrect. Since 4 and 2 were already eliminated as even numbers, this additionally removes 9. We're left with only four possible digits for our code: 1, 3, 5, and 7.

Step 2: Apply divisibility rule Clue 6 states that our code is not divisible by 5. A number is divisible by 5 if and only if its last digit is either 0 or 5. Since our number is NOT divisible by 5, we know the last digit cannot be 0 or 5. We already eliminated 0 as an even number, so this specifically tells us that position 3 cannot contain the digit 5.

Step 3: Sum constraint reveals exact digits Clue 5 tells us that the digits in odd positions (positions 1 and 3) sum to 12. Looking at our available digits {1, 3, 5, 7}, we need two that add to 12. Testing the combinations: 1+3=4, 1+5=6, 1+7=8, 3+5=8, 3+7=10, and 5+7=12. Only one pair works: 5 and 7 must be in positions 1 and 3. Since we already determined position 3 cannot be 5 (from the divisibility rule), position 3 must be 7 and position 1 must be 5.

Step 4: Determine the middle digit With positions 1 and 3 filled (5 and 7), we need to find position 2. Since we have no repeated digits (Clue 1), and we've already used 5 and 7, the middle digit must be either 1 or 3. Clue 2 provides the final piece: the test number 871 has exactly one correct digit in the wrong position. We know 7 is in our code (at position 3), and in this test it appears at position 2, which is indeed a "wrong position." If our middle digit were 1, then this test would have TWO correct digits (both 7 and 1), contradicting the clue. Therefore, the middle digit must be 3.

Step 1: No prime digits allowed Clue 4 immediately constrains our puzzle significantly - there are no prime digits anywhere in the code. Prime single digits are 2, 3, 5, and 7, so these are completely eliminated from all five positions. This leaves us with only the digits 0, 1, 4, 6, 8, and 9 to work with across the entire code.

Step 2: Positions 1 and 4 must sum to 17 Clue 3 tells us that the digits at positions 1 and 4 add up to 17. Given our limited digit pool (no primes), we need to find two single digits from {0, 1, 4, 6, 8, 9} that sum to 17. The only possible combination is 9 + 8 = 17. Therefore, positions 1 and 4 must contain 9 and 8 in some order - we just don't know which goes where yet.

Step 3: Working out the middle three positions We know the total sum is 31 (Clue 5), and positions 1 and 4 contribute 17 to this sum. This means positions 2, 3, and 5 must sum to 31 - 17 = 14. Additionally, Clue 6 tells us that the digits in positions 2 and 3 are identical. So we need to solve: X + X + Y = 14, where X is the repeated digit in positions 2 and 3, and Y is the digit in position 5.

Step 4: Finding valid values for the middle section Let's check what values of X could work. If X = 0, then 0 + 0 + Y = 14, meaning Y = 14, but there's no single digit 14. If X = 1, then 1 + 1 + Y = 14, so Y = 12, which also doesn't exist. If X = 4, then 4 + 4 + Y = 14, so Y = 6, and both 4 and 6 are available to us. If X = 6, then 6 + 6 + Y = 14, so Y = 2, but we can't use 2 since it's prime. The pattern continues - only X = 4, Y = 6 works with our constraints.

Step 5: Determining the final arrangement We now know our code contains the digits 9, 4, 4, 8, 6, but we need to determine whether position 1 is 9 or 8. Clue 2 provides the answer: the test number 91867 has exactly one digit in the correct position. Looking at our digits, we see that 9 appears in position 1 of this test. If our code has 9 in position 1, then this satisfies the "one correct position" requirement. Additionally, the test has 8 and 6 which are in our code but in different positions (8 is in position 4, not 3; 6 is in position 5, not 4), giving us the "2 digits correct but wrong positions." Everything checks out - our final code is 94486.