We have a 7-digit pickup code problem. The code consists of 7 digits, and we need to find these digits such that each pair of consecutive digits sums to specific values: 6, 5, 4, 3, 2, and 1 from left to right.
Now let's set up our system of equations. We have six equations: d1 plus d2 equals 6, d2 plus d3 equals 5, and so on until d6 plus d7 equals 1. Each digit must be between 0 and 9. This gives us a system of linear equations that we can solve step by step.
Let's solve this system step by step. From the first equation, d2 equals 6 minus d1. From the second equation, d3 equals 5 minus d2, which gives us d1 minus 1. Continuing this pattern, we can express all digits in terms of d1. We get d4 equals 5 minus d1, d5 equals d1 minus 2, d6 equals 4 minus d1, and d7 equals d1 minus 3.
Now we need to find the valid range for d1. Each digit must be between 0 and 9, so we set up inequalities for each expression. From d2 equals 6 minus d1, we get d1 must be at most 6. From d3 equals d1 minus 1, we get d1 must be at least 1. Continuing this process for all digits, we find that d1 must satisfy all constraints simultaneously. The intersection of all these constraints gives us d1 between 3 and 4, so d1 can only be 3 or 4.