Sir Birnie, a provincial governor in Australia during the colonial period, amassed great wealth while there. Later, back in England, he indulged two great passions: geometry and path building.
He also, it seems, considered his children to be incorrigible spendthrifts. Perhaps that was the reason for the rumor that beyond the fine fortune he left them, some of his wealth was hidden out of their reach. Further, he made two stipulations reflecting his concerns, further raising suspicions.
[Map: the letter "a" indicates the angle measure]
"First, I desire that the sealed envelope addressed 'To My First Grandchild' should be opened only by the first of my grandchildren to reach the age of maturity. The contents are to serve the benefit of all my grandchildren equally. Second, as my children should know, I own the lands between the road XQ and XR. Yes, I know that the maps call these roads Cotswold and Seeley, respectively, but I insist on these names. Seeley (XR) goes due east. There is also a path that I have built that is the angle bisector between those roads. No matter in what financial difficulties my spendthrift children find themselves, that bisector and all that is north of it must remain in trust for my grandchildren. If not, may you all be damned."
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His worries proved to be well-founded. His children proceeded to spend themselves into such debt that they found themselves forced to sell all the land south of the bisector line. They did, however, hold onto the road and the northern sliver, thus avoiding Sir Birnie's posthumous wrath. Sir Birnie's grandchildren were more industrious but uncomfortable in their relative poverty, so they lay great hope in the sealed letter.
The day finally arrived when the eldest grandchild, Johanna Haven, reached her age of maturity. She was a worthy descendant to Sir Birnie. She read mathematics at Oxford and was already renowned for her ability to prove topological theorems in eight dimensions.
The whole family gathered round her as she opened the sealed envelope. Besides a letter, there was a map and a smaller sealed envelope addressed 'To My First Great Grandchild'.
Johanna walked the XQ line but could not find Y. She walked the bisector and did find an old maple. The neighbor to the south of the bisector line, knowing the family lore and holding some inherited grudge against Sir Birnie, said that any visit to his property would cost 20,000 pounds. Johanna wondered whether it would be worth it to pay. On the one hand, the treasure could be worth a lot more. On the other, 20,000 pounds would put her substantially in debt. Her family urged her to borrow the money.
After a bit of sketching, she made a decision and as a result was able to find the treasure in her first attempt at digging. It cost her nothing. How did she do it?