16 Santa’s Delivery at the House Block

© Mar Curcó Iranzo, MATH+

Author: Mar Curcó Iranzo

Challenge

Santa is flying over Verdedam, a city famous for its gardens, to deliver presents. Each block of houses in the city is represented by a rectangular grid, where every cell is either a chimney, where Santa needs to drop presents, or a small garden, where elves are waiting to load more presents onto his sleigh. Loading and delivery of presents are represented by positive and negative integers, respectively. For example, a cell with the number 4 stands for a small garden, where 4 presents are loaded onto Santa’s sleigh. A cell with the number -4 stands for a chimney where Santa needs to drop 4 presents.

In Verdedam, the elves are highly organized. In every house block, they arrange the presents in the gardens in such a way that the following condition is satisfied: in every 4\times 3 or 3\times 4 rectangle of the corresponding grid, the sum of the entries is zero.

Figure 1: The corresponding grids of the first two house blocks. In every 4 \times 3 or 3 \times 4 rectangle of the grids, the sum of the entries is zero.

Santa arrives in Verdedam with an empty sleigh, flies from block to block and, within each block, always follows the same routine: he first visits all small gardens to load presents and only afterwards visits all chimneys, each exactly once. Considering the order in which Santa visits the blocks each year, his well-organized elves have of course distributed the presents such that at any chimney, Santa has enough presents in his sleigh to deliver the requested amount. The first block on his route, shown in Figure 1, is represented by a 6\times 7 grid whose entries sum to 1, so Santa leaves it with one present. The second block, also shown in Figure 1, is represented by a 3\times 5 grid whose entries sum to -1. Using the extra present from the previous block, Santa delivers all gifts and leaves this block with an empty sleigh.

As Santa continues his journey through Verdedam, strong winds force him to slightly change his route, and he reaches a different part of the city earlier than expected. There, he arrives at a third house block, represented by a 7\times 10 grid.

Figure 2: The grid corresponding to the third block of houses.

Santa knows that in the center of this block there are two small gardens where he loads 20 and 25 presents, as indicated in Figure 2. All 4\times 3 and 3\times 4 rectangles satisfy the usual condition.

Santa arrives at this third house block with an empty sleigh. Of course, the elves did not take this unforeseen event into consideration when distributing the gifts, so Santa may have or may have not enough presents to deliver the requested amounts in this block. How many presents will he fail to deliver, or how many presents will he have left over when he exits the block?

Hint: For the 7\times 10 grid in Figure 2, containing the fixed central entries 20 and 25, there exists at least one filling satisfying the condition that the sum of the entries in every 3\times 4 and every 4\times 3 rectangle is zero.

Possible answers

1. He will have failed to deliver 45 presents.
2. He will have failed to delive 25 presents.
3. He will have failed to deliver 20 presents.
4. He will have failed to deliver 5 presents.
5. In the block, there are exactly as many presents as as must be delivered.
6. He will have 5 presents left over in his sleigh.
7. He will have 20 presents left over in his sleigh.
8. He will have 25 presents left over in his sleigh.
9. He will have 45 presents left over in his sleigh.
10. It’s not possible to determine with the information we have.