Did you crack it?
Earlier today, I tossed an elegant little puzzle your way. Now we get to see what’s under the hood.
Nose to tail
The premise is simple. Brutally simple. You have a number N. It starts with a 4. Move that 4 to the very end. The new number is exactly one-quarter of the original.
In math terms?
N = 4[...]
And N ÷ 4 = [...]4
Find the smallest N that holds up.
The solution
Start small. Build outward.
First, try a two-digit number. Say N is 4?. The missing digit has to be 1. Why? Because a quarter of 40-49 is roughly 10-12. Specifically, a quarter of the 4 in the tens place is 1 in the new units place? No wait. 4x / 4 results in a number ending in 4?
Actually, just do the reverse multiplication.
4 times the shifted number equals the original.
So 4 * 1? must end in 4.
Only 4 * 6 ends in 4… wait no. Let’s stick to the forward logic. A quarter of 41 is 10.25. Too big.
Try N = 4?. 4 times the new number (?4 ) equals 4?.
If the last digit is 4, and we divide by 4, the last digit of the quotient is 1.
So ? is 1.
Check it. Is 14 a quarter of 41?
Nope. 41 / 4 is 10.75.
We need more digits.
Three digits. N = 41?.
Again. A quarter of 4... puts a 1 at the start of the quotient.
So the new number looks like 1?4.
Multiply back. 4 * 1?4 must equal 41?.
Look at the end. 4 * 4 = 16.
The last digit of N must be 6.
Is 416 / 4 equal to 164?
Do the math. 416 / 2 is 208. /2 again is 104.
Close. But not 164.
Try again. Four digits. N = 4..6.
We know the last digit is 6 and the first is 4.
What about the middle?
The number looks like 4..56? Let’s find that penultimate digit.
4 * [new number ending in 64] = [old number ending in ..6].
Actually, easier to track carries from the right.
Last digit: 6. 4*4=16, carry 1.
Second from right: We need the result to match.
4 * 1564 = 6256… wrong start.
4 * 1564 is roughly 6000. We need it to start with 4.
The shifted number is 1...64.
4 * 64 = 2556? No 4*64=256.
Ends in 56.
So N ends in 56.
Check: Is 4156 / 4 = 1564?
`4156 / 2 =
