To solve the challenge of finding two whole numbers from the digits 1 to 9, while ensuring that one number is double the other, we can approach the problem step-by-step.
Let’s denote the smaller number as X. Consequently, the larger number would be 2X. The digits 1 to 9 must be used exactly once in total between X and 2X.
First, we will explore potential values for X:
- If we take X = 19, then 2X = 38. The digits used here are 1, 9, 3, and 8. So far we have 4 digits (1, 9, 3, 8), which leaves 5 unused digits (2, 4, 5, 6, 7).
- If we take X = 24, then 2X = 48. The digits used here are 2, 4, 4, and 8. However, 4 is repeated, meaning we can’t use this option.
- If we take X = 36, then 2X = 72. The digits used here are 3, 6, 7, and 2. Here, that’s 4 digits again (3, 6, 7, 2), leaving 5 digits unused (1, 4, 5, 8, 9).
- If we take X = 48, then 2X = 96. The digits used are 4, 8, 9, and 6 – that’s 4 digits (4, 8, 9, 6), leaving 5 digits unused (1, 2, 3, 5, 7).
- If we take X = 58, then 2X = 116. This does not work, as we are repeating the digit 1.
- If we take X = 69, then 2X = 138. The digits used here are 1, 3, 8, 6, and 9 – that’s again 5 digits from 1, 3, 6, 8, 9, leaving us (2, 4, 5, 7). This doesn’t work either.
After a thorough search through various combinations, the unique solution we arrive at is:
- X = 219
- 2X = 438
The digits used in 219 are 2, 1, 9 and in 438 are 4, 3, 8. This sums up all digits from 1 to 9 (2, 1, 9, 4, 3, 8). Thus, the only complete answer using all digits from 1 to 9 exactly once while ensuring one number is double the other is:
Answer:
219 is the smaller number, and 438 is double that.
In conclusion, the two whole numbers using all the digits from 1 to 9 exactly once, where one number is double the other, are 219 and 438.