A Puzzle A Day: 2026-04-25

Correct: 1

Let 'N' be the number of ingredients. We need to find the smallest 'N' (where N < 6) that satisfies all conditions. Let's analyze the conditions for each possible value of N, starting from the smallest: * **Condition #3:** 'If the quantities are listed in ascending order, the difference between any two consecutive quantities is always 2.' This is the trickiest condition to interpret. * **Case N = 1:** * If there's only one ingredient, say with quantity `x_1`. There are no 'two consecutive quantities'. Therefore, the condition 'the difference between any two consecutive quantities is always 2' is vacuously true – there are no such pairs that could fail this condition. * Condition #1: `x_1` must be a whole number greater than 0. The smallest such number is 1. * Condition #4: The sum of all quantities (`x_1`) must be a perfect square. If `x_1 = 1`, then the sum is 1, which is `1^2` (a perfect square). * Condition #5: `N=1` is less than 6. * All conditions are satisfied for `N=1` with the quantity `[1]`. Thus, 1 is a possible number of ingredients. * **Case N = 2:** * Let the quantities be `x_1, x_2` in ascending order. Condition #3 means `x_2 - x_1 = 2`. * Condition #1: Smallest `x_1` is 1. So quantities are `[1, 3]`. * Condition #4: The sum is `1 + 3 = 4`. `4` is a perfect square (`2^2`). * Condition #5: `N=2` is less than 6. * All conditions are satisfied for `N=2` with quantities `[1, 3]`. Thus, 2 is a possible number of ingredients. Since the question asks for the *smallest possible number* of ingredients, and we found that `N=1` satisfies all conditions, the answer is 1. The common mistake is to assume that 'consecutive quantities' implies there must be at least two quantities, making `N=2` the smallest.