# Find the 3 Item Amounts

Last week the store's receipt printer removed lines randomly from each item amount.

A technician repaired the printer, but now it adds lines randomly to each item amount!

The cashier told me that for receipts with 3 items (like mine below):

1. The total is always correct.
2. It adds more lines to the digits in the 2nd item than to the digits in the 1st item, and more lines to the digits in the 3rd item than to the digits in the 2nd item.
3. It always adds 14 lines in total.

How much did I pay for each item?

The accepted answer will articulate all logic steps.

Here is the receipt:

• Is it possible that the machine adds zero lines to an item amount? Jun 28, 2022 at 15:36
• @LeppyR64 Good question. Yes, that is possible, but I'm not saying if it did or didn't happen for the receipt in the puzzle.
– JLee
Jun 28, 2022 at 16:22

Starting in the hundredths place (the rightmost column):

There’s a 5 at the top and the 9 below it cannot have been a 0, so the total is 15.

Therefore we need to change the 9 and 8 into a pair of digits that sum to 10.

Possible digits for this column (reading from top to bottom, and keeping track of how many lines are changed for each) are then 591 (005), 573 (032), 555 (012), 537 (014), or 519 (041).

Moving to the tenths place:

We’re carrying a 1 from the column to the right.

There’s a 1 at the top, so the total of the column must be 11. We need to change the 8 and 9 to digits that sum to 9.

Possible digits for this column are then 181 (004), 163 (011), 145 (031), 127 (023), or 109 (010).

Moving to the ones place:

We’re carrying a 1 from the column to the right. The total of this column must have been 13.

Possible digits for this column are then 067 (002), 157 (412), or 751 (313).

Moving to the tens place:

We’re carrying a 1 from the column to the right. The total of this column must have been 10.

The only way to accomplish this is if the digits are 514 (010).

Now focusing on the number of lines added:

If we only add 2 lines in the ones place, then the total number of lines added in the whole puzzle could have only been 13 at most. So 7 lines were added in the ones place.

If 4 lines were added in the top item, then it could not have added more in the 2nd and 3rd items, respectively, without exceeding 14. So the ones place is 751.

Since we’ve added a total of 8 lines in the dollars columns, we need another 6 lines added in the cents. This is only possible if 1 line is added in the tenths place, and 5 lines are added in the hundredths place. The tenths place is therefore 109.

We’ve added 3 to the top item and only 2 to the middle item, so we need at least 1 more in the middle item. We’ve added 3 to the bottom item, so we also need at least 2 more in the bottom item. This is only possible if the hundredths column is 537.

So the solution is:

The prices of the three items are 57.15, 15.03, and 41.97.