Hints:

Try pairing the numbers strategically:

1. Start by pairing 1 & 2 and 3 & 4. This results in new values and leaves 1, 1, 5, which can be further reduced to just 5.

2. Alternatively, pair 2 & 3 and 4 & 5. This leaves 1, 1, 1, which can be reduced to 1.

Question:
How can you determine whether 0 or 1 will be the lowest possible value that can be achieved when applying such operations to a given range of numbers?

Solution:

This puzzle is similar in spirit to the classic problem where you place addition or subtraction signs between the numbers 1 to 5 — for example:
1 + 2 + 3 - 4 + 5 or 1 + 2 - 3 + 4 + 5.
The nice thing about our version is that only nonnegative numbers are involved, since we’re using differences (absolute values), which ensures the final result is never less than 0.

Let’s explore how to determine which values are possible final results when performing such pairwise operations on the numbers from 1 to n. We’ll consider the range 1 to 5 first.

Observations:

1. All differences are nonnegative, so the minimum possible result is 0.

2. The values you get in intermediate steps are formed by taking absolute differences between two numbers—these may be original values or derived ones.

3. The maximum number that can appear during these operations is simply the maximum in the original list (in this case, 5), so no final result will exceed this.

Strategy: Parity Analysis (Evens and Odds)

Let’s look at the number of odd numbers in the initial list, as this gives a key insight.

For the set {1, 2, 3, 4, 5}, the odd numbers are 1, 3, and 5 — a total of three, which is odd.

Now analyze what happens when we take differences:

• Even – Even = Even → parity stays the same.

• Odd – Even = Odd (or vice versa) → parity stays the same.

• Odd – Odd = Even → reduces the number of odd numbers by two.

This means:

• The total number of odd numbers either stays the same or decreases by 2 with each operation.

• Therefore, if you start with an odd number of odd numbers, you will always end with one odd number.

• If you start with an even number of odd numbers, you can eventually end with zero odd numbers.

Applying This to 1–5:

• Start: 3 odd numbers → odd count.

• So, the final result must be odd: possible values are 1, 3, or 5.

• With some trial runs, you can confirm that each of 1, 3, and 5 is achievable.

Extending to Other Ranges:

• 1 to 6 → Odd numbers = {1, 3, 5} → 3 total → odd → final result must also be odd (1, 3, or 5).

• 1 to 7 → Odd numbers = {1, 3, 5, 7} → 4 total → even → final result must be even (0, 2, 4, 6).

Each of these values can also be achieved with the right sequence of operations.

Final Rule:

To determine whether the lowest possible final result is 0 or 1, simply:

• Count the number of odd numbers in the original range.

  • If it’s even, 0 is possible.

  • If it’s odd, the lowest possible result is 1.

Exploration:

It is straightforward to determine when a result of 0 is possible. Start by pairing up consecutive numbers from the top and calculating their differences. This process will yield a collection of 1’s. If you have an even number of 1’s, you can obtain a 0; if you have an odd number, you will get a 1. This method aligns with identifying whether you started with an even or odd number of odd numbers. For example, consider the numbers from 1 to 7: – Pair them as follows: (7, 6), (5, 4), (3, 2) and leave 1 alone. – The differences are 1, 1, and 1, which gives you a list of 1’s: 1, 1, 1, 1. – Next, pair those 1’s: (1, 1), (1, 1) to yield 0, 0. – Finally, this results in a final output of 0. To check if the top number is possible, follow a similar process. Pair up consecutive numbers starting from the top while excluding the top number. Calculate the differences of these pairs, resulting in a list of 1’s alongside the top number. Reduce the list of 1’s into either a single 0 or a single 1. Then, find the difference between that result and the top number!